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#### Page No 46:

#### Question 1:

#### Answer:

Yes, heat is a conserved quantity. Suppose two bodies A and B are at different temperatures, wherein A is higher and B is lower. When they are brought in contact, the heat given by A is equal to the heat gained by B. Thus, the heat is conserved in the system.

Here, we have ignored the heat exchanged with the surroundings. If we consider the surroundings, the heat of the system will be conserved.

#### Page No 46:

#### Question 2:

Yes, heat is a conserved quantity. Suppose two bodies A and B are at different temperatures, wherein A is higher and B is lower. When they are brought in contact, the heat given by A is equal to the heat gained by B. Thus, the heat is conserved in the system.

Here, we have ignored the heat exchanged with the surroundings. If we consider the surroundings, the heat of the system will be conserved.

#### Answer:

One calorie is defined as the heat required to raise the temperature of 1 g of water from 14.5^{o} C to 15.5^{o} C. Also, exactly 4.186 J of work is to be done on heating 1 g of water from 14.5^{o} C to 15.5^{o} C. So, we cannot round-off 4.186 J to 4 J as it will give significant difference in work done required to raise the temperature and heat required to raise the temperature.

Since work and heat are equivalent here, so on taking 1 calorie = 4 J, the validity of the equivalence of work and heat will get defied. Thus, just to make conversion easy, 1 calorie cannot be taken equal to 4 J.

#### Page No 46:

#### Question 3:

One calorie is defined as the heat required to raise the temperature of 1 g of water from 14.5^{o} C to 15.5^{o} C. Also, exactly 4.186 J of work is to be done on heating 1 g of water from 14.5^{o} C to 15.5^{o} C. So, we cannot round-off 4.186 J to 4 J as it will give significant difference in work done required to raise the temperature and heat required to raise the temperature.

Since work and heat are equivalent here, so on taking 1 calorie = 4 J, the validity of the equivalence of work and heat will get defied. Thus, just to make conversion easy, 1 calorie cannot be taken equal to 4 J.

#### Answer:

A calorimeter is kept in a wooden box to insulate it thermally from the surroundings because in order to determine the specific heat capacity, the total heat transferred must be known. Heat must not be exchanged with the surroundings, otherwise the principle of calorimeter would not stand valid.

#### Page No 46:

#### Question 4:

A calorimeter is kept in a wooden box to insulate it thermally from the surroundings because in order to determine the specific heat capacity, the total heat transferred must be known. Heat must not be exchanged with the surroundings, otherwise the principle of calorimeter would not stand valid.

#### Answer:

Yes, heat of the two objects taken together remains constant. If no heat is lost to the surroundings, the heat of the two bodies taken together actually remains conserved.

#### Page No 46:

#### Question 5:

Yes, heat of the two objects taken together remains constant. If no heat is lost to the surroundings, the heat of the two bodies taken together actually remains conserved.

#### Answer:

The inlet is near the top and the outlet is near the bottom because there is a loss of heat from the steam as it passes through the chamber. As the steam loses heat, a part of it condenses back to water and the cold steam gets denser and moves down towards the bottom.

But when done the other way round, the used steam does not pass through the chamber correctly and gets mixed up. This can result in discrepancy in the desired results.

#### Page No 46:

#### Question 6:

The inlet is near the top and the outlet is near the bottom because there is a loss of heat from the steam as it passes through the chamber. As the steam loses heat, a part of it condenses back to water and the cold steam gets denser and moves down towards the bottom.

But when done the other way round, the used steam does not pass through the chamber correctly and gets mixed up. This can result in discrepancy in the desired results.

#### Answer:

When a solid melts or a liquid boils, the heat supplied is actually used to break the bond forces between the molecules and bring them apart till the body changes its state completely. Thus, the energy is transferred to the molecules as kinetic energy and the temperature of body remains constant in the process.

#### Page No 46:

#### Question 7:

When a solid melts or a liquid boils, the heat supplied is actually used to break the bond forces between the molecules and bring them apart till the body changes its state completely. Thus, the energy is transferred to the molecules as kinetic energy and the temperature of body remains constant in the process.

#### Answer:

(a) The specific heat capacity of melting ice is 0.50 cal/g-^{o}C or 2093 J/kg-K.

(b) The specific heat capacity of boiling water is 0.46 cal/g-^{o}C or 1926 J/kg-K.

#### Page No 46:

#### Question 8:

(a) The specific heat capacity of melting ice is 0.50 cal/g-^{o}C or 2093 J/kg-K.

(b) The specific heat capacity of boiling water is 0.46 cal/g-^{o}C or 1926 J/kg-K.

#### Answer:

Steam has greater energy than boiling water due to latent heat of vapourisation. The internal energy of the vapour at 100^{ o}C is greater than the internal energy of the boiling water at the same temperature. Thus, steam burns the skin more severely than boiling water.

#### Page No 46:

#### Question 9:

Steam has greater energy than boiling water due to latent heat of vapourisation. The internal energy of the vapour at 100^{ o}C is greater than the internal energy of the boiling water at the same temperature. Thus, steam burns the skin more severely than boiling water.

#### Answer:

The atmospheric temperature in the cities on the sea-coast changes very little due to the following reasons:

1) The specific heat capacity of water is four times greater than land. This means water heats and cools more slowly than land. Also, once the ocean has warmed, it gives up its heat much slower than the land.

2) The moisture content of the air over coastal locations is very high. Moisture in the form of water vapour is the predominate greenhouse gas in the troposphere. Water vapour prevents the loss of heat into space at night. Thus, the temperature during day and night is almost the same.

#### Page No 46:

#### Question 10:

The atmospheric temperature in the cities on the sea-coast changes very little due to the following reasons:

1) The specific heat capacity of water is four times greater than land. This means water heats and cools more slowly than land. Also, once the ocean has warmed, it gives up its heat much slower than the land.

2) The moisture content of the air over coastal locations is very high. Moisture in the form of water vapour is the predominate greenhouse gas in the troposphere. Water vapour prevents the loss of heat into space at night. Thus, the temperature during day and night is almost the same.

#### Answer:

The thermometer bulb should have large heat capacity because if it has lower heat capacity, it may expand at high temperatures resulting in false readings. Large heat capacity of the thermometer bulb ensures the correct reading of temperature by restricting the bulb from expanding.

#### Page No 46:

#### Question 1:

The thermometer bulb should have large heat capacity because if it has lower heat capacity, it may expand at high temperatures resulting in false readings. Large heat capacity of the thermometer bulb ensures the correct reading of temperature by restricting the bulb from expanding.

#### Answer:

(d) the material of the body

Heat capacity of a body is due to their material properties. Due to different molecular structures, different bodies have a different capacity to absorb heat. Therefore, specific heat of a body depends on the material of the body.

#### Page No 46:

#### Question 2:

(d) the material of the body

Heat capacity of a body is due to their material properties. Due to different molecular structures, different bodies have a different capacity to absorb heat. Therefore, specific heat of a body depends on the material of the body.

#### Answer:

(a) kg

Since water equivalent of a body is the mass of the water having the same heat capacity as the given body, the water equivalent is measured in kilogram.

#### Page No 46:

#### Question 3:

(a) kg

Since water equivalent of a body is the mass of the water having the same heat capacity as the given body, the water equivalent is measured in kilogram.

#### Answer:

(d) is undefined for some time and then becomes nearly constant

When hot liquid is mixed with cold liquid, the molecules collide and transfer heat. When heat is transferred, the temperature is undefined. Once the heat energy is shared by the molecules, the system reaches equilibrium and the temperature becomes nearly constant.

#### Page No 46:

#### Question 4:

(d) is undefined for some time and then becomes nearly constant

When hot liquid is mixed with cold liquid, the molecules collide and transfer heat. When heat is transferred, the temperature is undefined. Once the heat energy is shared by the molecules, the system reaches equilibrium and the temperature becomes nearly constant.

#### Answer:

(d) Joule and calorie

One calorie is defined as the amount of heat needed to raise the temperature of 1 g of water from 14.5^{o} to 15.5^{o} at the pressure of 1 atm. Heat is a form of energy and the unit of energy is joule.

Therefore, joule and calorie represent energy.

#### Page No 46:

#### Question 5:

(d) Joule and calorie

One calorie is defined as the amount of heat needed to raise the temperature of 1 g of water from 14.5^{o} to 15.5^{o} at the pressure of 1 atm. Heat is a form of energy and the unit of energy is joule.

Therefore, joule and calorie represent energy.

#### Answer:

(c) Heat and Work

As work done in raising temperature of a body is actually the heat supplied to the body, heat and work may be represented in the same unit.

#### Page No 46:

#### Question 6:

(c) Heat and Work

As work done in raising temperature of a body is actually the heat supplied to the body, heat and work may be represented in the same unit.

#### Answer:

(c) Total internal energy of the two bodies

When two bodies at different temperatures are mixed in the calorimeter, heat flows from one body to the other due to the temperature difference. This results in change in the internal energy of the individual bodies. There is no exchange of heat with the surrounding in the calorimeter. Thus, the total internal energy of the bodies remain conserved as no external work is done on them.

#### Page No 46:

#### Question 7:

(c) Total internal energy of the two bodies

When two bodies at different temperatures are mixed in the calorimeter, heat flows from one body to the other due to the temperature difference. This results in change in the internal energy of the individual bodies. There is no exchange of heat with the surrounding in the calorimeter. Thus, the total internal energy of the bodies remain conserved as no external work is done on them.

#### Answer:

(d) is dimensionless

If the mechanical work done *(W)* produces the same temperature change as heat *(H),* then the mechanical equivalent of heat (*J) *is equal to *W*/*H. *Thus,*J* = *W*/*H*

Since the unit of work and heat is the same, mechanical equivalent of heat is dimensionless.

#### Page No 46:

#### Question 1:

(d) is dimensionless

If the mechanical work done *(W)* produces the same temperature change as heat *(H),* then the mechanical equivalent of heat (*J) *is equal to *W*/*H. *Thus,*J* = *W*/*H*

Since the unit of work and heat is the same, mechanical equivalent of heat is dimensionless.

#### Answer:

(c) the mass of the body

(d) the material of the body

The bigger the body, the larger is its capacity to absorb heat. Therefore, the heat capacity of a body depends on the mass of the body. Also, different bodies have different heat capacities due to their material properties, i.e. due to their molecular structure, the heat capacity of a body depends on the material of the body.

#### Page No 46:

#### Question 2:

(c) the mass of the body

(d) the material of the body

The bigger the body, the larger is its capacity to absorb heat. Therefore, the heat capacity of a body depends on the mass of the body. Also, different bodies have different heat capacities due to their material properties, i.e. due to their molecular structure, the heat capacity of a body depends on the material of the body.

#### Answer:

(c) depends on the molecular weight of the body

Specific heat capacity of a body, $s=\frac{Q}{m\Delta \theta}$

Here,*Q* = Heat supplied*m* = Mass of body

Δ*θ** *= Change in temperature

Molar heat capacity of a body, $C=\frac{Q}{n\Delta \theta}\phantom{\rule{0ex}{0ex}}$

Here,*Q* = Heat supplied*n* = Number of moles

Δ*θ* = Change in temperature

∴ The ratio of the specific heat capacity and molar heat capacity is given by

$\frac{s}{C}=\frac{\frac{Q}{m\Delta \theta}}{{\displaystyle \frac{Q}{n\Delta \theta}}}=\frac{n}{m}=\frac{n}{nM}=\frac{1}{M}\phantom{\rule{0ex}{0ex}}$

Here,*M* = Molar mass related to number of moles*m* = Mass

As the value of *M *is different for different bodies of different composition, the ratio cannot be a universal constant.

Also, the ratio is independent of the mass of the body.

The ratio of the specific heat and molar heat capacity depends on the molecular weight of the body.

Clearly, the unit of molecular weight is kg/mole. So, the ratio that depends only on the molecular weight cannot be dimensionless.

#### Page No 46:

#### Question 3:

(c) depends on the molecular weight of the body

Specific heat capacity of a body, $s=\frac{Q}{m\Delta \theta}$

Here,*Q* = Heat supplied*m* = Mass of body

Δ*θ** *= Change in temperature

Molar heat capacity of a body, $C=\frac{Q}{n\Delta \theta}\phantom{\rule{0ex}{0ex}}$

Here,*Q* = Heat supplied*n* = Number of moles

Δ*θ* = Change in temperature

∴ The ratio of the specific heat capacity and molar heat capacity is given by

$\frac{s}{C}=\frac{\frac{Q}{m\Delta \theta}}{{\displaystyle \frac{Q}{n\Delta \theta}}}=\frac{n}{m}=\frac{n}{nM}=\frac{1}{M}\phantom{\rule{0ex}{0ex}}$

Here,*M* = Molar mass related to number of moles*m* = Mass

As the value of *M *is different for different bodies of different composition, the ratio cannot be a universal constant.

Also, the ratio is independent of the mass of the body.

The ratio of the specific heat and molar heat capacity depends on the molecular weight of the body.

Clearly, the unit of molecular weight is kg/mole. So, the ratio that depends only on the molecular weight cannot be dimensionless.

#### Answer:

(b) may increase

(c) may remain constant

When heat is supplied to a solid, it is used up either to increase the temperature of the body or to change its state from one form to another by breaking the bonds between the molecules (without raising the temperature).

When heat is supplied to the solid, the internal energy of the solid increases, so the temperature does not decrease.

#### Page No 46:

#### Question 4:

(b) may increase

(c) may remain constant

When heat is supplied to a solid, it is used up either to increase the temperature of the body or to change its state from one form to another by breaking the bonds between the molecules (without raising the temperature).

When heat is supplied to the solid, the internal energy of the solid increases, so the temperature does not decrease.

#### Answer:

(a) heat may have been supplied to the body

(b) heat may have been extracted from the body

If there is no temperature change in a solid object, there is a possibility that the heat might have been supplied to the body that was used up in breaking the bond of the molecules, changing the state of the solid. This is why the temperature of the solid remians constant. Similar is the case when the heat is extracted from the body to change its state.

Since there is a possibility of supplying or extracting heat from the solid, we cannot say that heat is not supplied to the solid or is not extracted from the solid.

#### Page No 46:

#### Question 5:

(a) heat may have been supplied to the body

(b) heat may have been extracted from the body

If there is no temperature change in a solid object, there is a possibility that the heat might have been supplied to the body that was used up in breaking the bond of the molecules, changing the state of the solid. This is why the temperature of the solid remians constant. Similar is the case when the heat is extracted from the body to change its state.

Since there is a possibility of supplying or extracting heat from the solid, we cannot say that heat is not supplied to the solid or is not extracted from the solid.

#### Answer:

(c) heat may have been supplied to it

(d) work may have been done on it

If the temperature of an object rises in a period, then there are two possibilities. The heat may have been supplied to it, leading to an increase of the internal energy of the object. That will increase the temperature of the body.

The second possibility is that some work may have been done on it, again leading to an increase of the internal energy of the body. That will also increase the temperature of the body.

#### Page No 46:

#### Question 6:

(c) heat may have been supplied to it

(d) work may have been done on it

If the temperature of an object rises in a period, then there are two possibilities. The heat may have been supplied to it, leading to an increase of the internal energy of the object. That will increase the temperature of the body.

The second possibility is that some work may have been done on it, again leading to an increase of the internal energy of the body. That will also increase the temperature of the body.

#### Answer:

(c) the temperature of a body can be increased by doing work on it

According to the statement "heat and work are equivalent", heat supplied to the body increases its temperature. Similarly, work done on the body also increases its temperature.

For example: If work is done on rubbing the hands against each other, the temperature of the hands increases. So, we can say that heat and work are equivalent.

When heat is supplied to a body, we do not do work on it. When we are doing work on a body, it does not mean we are supplying heat to the body. Also, a body at rest cannot be set in motion along a line by supplying heat to it. So, these statements do not justify the equivalence of heat and work.

#### Page No 47:

#### Question 1:

(c) the temperature of a body can be increased by doing work on it

According to the statement "heat and work are equivalent", heat supplied to the body increases its temperature. Similarly, work done on the body also increases its temperature.

For example: If work is done on rubbing the hands against each other, the temperature of the hands increases. So, we can say that heat and work are equivalent.

When heat is supplied to a body, we do not do work on it. When we are doing work on a body, it does not mean we are supplying heat to the body. Also, a body at rest cannot be set in motion along a line by supplying heat to it. So, these statements do not justify the equivalence of heat and work.

#### Answer:

Given:

Mass of aluminium = 0.5 kg

Mass of water = 0.2 kg

Mass of iron = 0.2 kg

Specific heat of aluminium = 910 J kg^{−1} K^{−1}

Specific heat of iron = 470 J kg^{−1} K^{−1}

Specific heat of water = 4200 J kg^{−1} K^{−1}

Let the equilibrium temperature of the mixture be *T*.

Temperature of aluminium and water = 20°C = 273+20 = 293 K

Temperature of iron = 100°C = 273 + 100 = 373 K

Heat lost by iron, *H*_{1} = 0.2 × 470 × (373 − *T*)

Heat gained by water = 0.2 × 4200 × (*T* − 293)

Heat gained by iron = 0.5 × 910 × (*T*−293)

Total heat gained by water and iron, *H*_{2} = 0.5 × 910 (*T*−293) + 0.2 × 4200 × (*T* − 293)

*H*_{2} = (*T* − 293) [0.5 × 910 + 0.2 × 4200]

We know,

Heat gain = Heat lost

⇒ (*T* − 293) [0.5 × 910 + 0.2 × 4200] = 0.2 × 470 × (373 −*T*)

⇒ (*T* − 293) (455 + 840) = 94 (373 − *T*)

$\Rightarrow \left(T-293\right)\frac{1295}{94}=\left(373-T\right)\phantom{\rule{0ex}{0ex}}\left(T-293\right)\times 14=\left(373-T\right)$

⇒ 14 *T* − 293 × 14 = 373 − *T*

⇒ 15 *T* = 373 + 4102 = 4475

$\Rightarrow T=\frac{4475}{15}=298.33\mathrm{K}\approx 298\mathrm{K}$

∴ *T* = (298 − 273)°C = 25°C

∴ Final temperature = 25° C

#### Page No 47:

#### Question 2:

Given:

Mass of aluminium = 0.5 kg

Mass of water = 0.2 kg

Mass of iron = 0.2 kg

Specific heat of aluminium = 910 J kg^{−1} K^{−1}

Specific heat of iron = 470 J kg^{−1} K^{−1}

Specific heat of water = 4200 J kg^{−1} K^{−1}

Let the equilibrium temperature of the mixture be *T*.

Temperature of aluminium and water = 20°C = 273+20 = 293 K

Temperature of iron = 100°C = 273 + 100 = 373 K

Heat lost by iron, *H*_{1} = 0.2 × 470 × (373 − *T*)

Heat gained by water = 0.2 × 4200 × (*T* − 293)

Heat gained by iron = 0.5 × 910 × (*T*−293)

Total heat gained by water and iron, *H*_{2} = 0.5 × 910 (*T*−293) + 0.2 × 4200 × (*T* − 293)

*H*_{2} = (*T* − 293) [0.5 × 910 + 0.2 × 4200]

We know,

Heat gain = Heat lost

⇒ (*T* − 293) [0.5 × 910 + 0.2 × 4200] = 0.2 × 470 × (373 −*T*)

⇒ (*T* − 293) (455 + 840) = 94 (373 − *T*)

$\Rightarrow \left(T-293\right)\frac{1295}{94}=\left(373-T\right)\phantom{\rule{0ex}{0ex}}\left(T-293\right)\times 14=\left(373-T\right)$

⇒ 14 *T* − 293 × 14 = 373 − *T*

⇒ 15 *T* = 373 + 4102 = 4475

$\Rightarrow T=\frac{4475}{15}=298.33\mathrm{K}\approx 298\mathrm{K}$

∴ *T* = (298 − 273)°C = 25°C

∴ Final temperature = 25° C

#### Answer:

Given:

Mass of iron = 100 g

Water equivalent of calorimeter = 10 g

Mass of water = 240 gm

Let the temperature of surface be $\theta $ °C.

Specific heat capacity of iron = 470 J kg^{−1} °C^{−1}

Total heat gained = Total heat lost

$\Rightarrow \frac{100}{1000}\times 470\times \left(\theta -60\xb0\right)=\frac{\left(240+10\right)}{1000}\times 4200\times \left(60-20\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 47\theta -47\times 60=25\times 42\times 40\phantom{\rule{0ex}{0ex}}\Rightarrow \theta \mathit{}=\frac{42000+2820}{47}=\frac{44820}{47}\phantom{\rule{0ex}{0ex}}=953.61\xb0\mathrm{C}$

#### Page No 47:

#### Question 3:

Given:

Mass of iron = 100 g

Water equivalent of calorimeter = 10 g

Mass of water = 240 gm

Let the temperature of surface be $\theta $ °C.

Specific heat capacity of iron = 470 J kg^{−1} °C^{−1}

Total heat gained = Total heat lost

$\Rightarrow \frac{100}{1000}\times 470\times \left(\theta -60\xb0\right)=\frac{\left(240+10\right)}{1000}\times 4200\times \left(60-20\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 47\theta -47\times 60=25\times 42\times 40\phantom{\rule{0ex}{0ex}}\Rightarrow \theta \mathit{}=\frac{42000+2820}{47}=\frac{44820}{47}\phantom{\rule{0ex}{0ex}}=953.61\xb0\mathrm{C}$

#### Answer:

Given:

Temperature of A = 12°C

Temperature of B = 19°C

Temperature of C = 28°C

Temperature of mixture of A and B = 16°C

Temperature of mixture of B and C = 23°C

Let the mass of the mixtures be *M *and the specific heat capacities of the liquids A, B and C be *C*_{A}, *C*_{B}, and *C*_{C}, respectively.

According to the principle of calorimetry, when A and B are mixed, we get

Heat gained by Liquid A = Heat lost by liquid B

⇒ *MC*_{A} (16 − 12) = *MC*_{B} (19 − 16)

⇒ 4*MC*_{A} = 3 *MC*_{B}

$\Rightarrow M{C}_{\mathrm{A}}=\left(\frac{3}{4}\right)M{C}_{\mathrm{B}}$ ...(1)

When B and C are mixed:

Heat gained by liquid B = Heat lost by liquid C

⇒*MC*_{B} (23 − 19) = *MC*_{C} (28 − 23)

⇒ 4*MC*_{B} = 5 *MC*_{C}

$\Rightarrow {\mathrm{M}}_{\mathrm{CC}}=\left(\frac{4}{5}\right){\mathrm{M}}_{\mathrm{CB}}$ ...(2)

When A and C are mixed:

Let the temperature of the mixture be *T.* Then,

Heat gained by liquid A = Heat lost by liquid C

⇒ *MC*_{A} (*T* − 12) = *MC*_{C} (28 − *T*)

Using the values of *MC*_{A }and *MC*_{C}, we get

$\Rightarrow \left(\frac{3}{4}\right)M{C}_{\mathrm{B}}\left(T-12\right)=\left(\frac{4}{5}\right)M{C}_{\mathrm{B}}\left(28-T\right)\left[\mathrm{From}\mathrm{eqs}.\left(1\right)\mathrm{and}\left(2\right)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow \left(\frac{3}{4}\right)\left(T-12\right)=\left(\frac{4}{5}\right)\left(28-T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \left(3\times 5\right)\left(T-12\right)=\left(4\times 4\right)\left(28-T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 15T-180=448-16T\phantom{\rule{0ex}{0ex}}\Rightarrow 31T=628\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{628}{31}=20.253\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow T=20.3\xb0\mathrm{C}$

#### Page No 47:

#### Question 4:

Given:

Temperature of A = 12°C

Temperature of B = 19°C

Temperature of C = 28°C

Temperature of mixture of A and B = 16°C

Temperature of mixture of B and C = 23°C

Let the mass of the mixtures be *M *and the specific heat capacities of the liquids A, B and C be *C*_{A}, *C*_{B}, and *C*_{C}, respectively.

According to the principle of calorimetry, when A and B are mixed, we get

Heat gained by Liquid A = Heat lost by liquid B

⇒ *MC*_{A} (16 − 12) = *MC*_{B} (19 − 16)

⇒ 4*MC*_{A} = 3 *MC*_{B}

$\Rightarrow M{C}_{\mathrm{A}}=\left(\frac{3}{4}\right)M{C}_{\mathrm{B}}$ ...(1)

When B and C are mixed:

Heat gained by liquid B = Heat lost by liquid C

⇒*MC*_{B} (23 − 19) = *MC*_{C} (28 − 23)

⇒ 4*MC*_{B} = 5 *MC*_{C}

$\Rightarrow {\mathrm{M}}_{\mathrm{CC}}=\left(\frac{4}{5}\right){\mathrm{M}}_{\mathrm{CB}}$ ...(2)

When A and C are mixed:

Let the temperature of the mixture be *T.* Then,

Heat gained by liquid A = Heat lost by liquid C

⇒ *MC*_{A} (*T* − 12) = *MC*_{C} (28 − *T*)

Using the values of *MC*_{A }and *MC*_{C}, we get

$\Rightarrow \left(\frac{3}{4}\right)M{C}_{\mathrm{B}}\left(T-12\right)=\left(\frac{4}{5}\right)M{C}_{\mathrm{B}}\left(28-T\right)\left[\mathrm{From}\mathrm{eqs}.\left(1\right)\mathrm{and}\left(2\right)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow \left(\frac{3}{4}\right)\left(T-12\right)=\left(\frac{4}{5}\right)\left(28-T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \left(3\times 5\right)\left(T-12\right)=\left(4\times 4\right)\left(28-T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 15T-180=448-16T\phantom{\rule{0ex}{0ex}}\Rightarrow 31T=628\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{628}{31}=20.253\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow T=20.3\xb0\mathrm{C}$

#### Answer:

(a)

Given:

Number of ice cubes = 4

Volume of each ice cube = (2 × 2 × 2) = 8 cm^{3}

Density of ice = 900 kg m^{−3}^{}

Total mass of ice, *m*_{i} = (4×8 ×10^{−6} ×900) = 288×10^{−4} kg

Latent heat of fusion of ice, *L*_{i} = 3.4 × 10^{5} J kg^{−1}

Density of the drink = 1000 kg m^{−3}

Volume of the drink = 200 ml

Mass of the drink = (200×10^{−6})×1000 kg

Let us first check the heat released when temperature of 200 ml changes from 10^{o}C to 0^{o}C.*H*_{w} = (200×10^{−6})×1000×4200×(10−0) = 8400 J

Heat required to change four 8 cm^{3} ice cubes into water (*H*_{i}) = *m*_{i}*L*_{i} = (288×10^{−4})×(3.4×10^{5}) = 9792 J

Since the heat required for melting the four cubes of the ice is greater than the heat released by water ( *H*_{i} > *H*_{w} ), some ice will remain solid and there will be equilibrium between

ice and water. Thus, the thermal equilibrium will be attained at 0^{o} C.

(b)

Equilibrium temperature of the cube and the drink = 0°C

Let *M* be the mass of melted ice.

Heat released when temperature of 200 ml changes from 10^{o}C to 0^{o}C is given by*H*_{w} = (200×10^{−6})×1000×4200×(10−0) = 8400 J

Thus,*M*×(3.4×10^{5}) = 8400 J

Therefore,*M *= 0.0247 Kg = 25 g

#### Page No 47:

#### Question 5:

(a)

Given:

Number of ice cubes = 4

Volume of each ice cube = (2 × 2 × 2) = 8 cm^{3}

Density of ice = 900 kg m^{−3}^{}

Total mass of ice, *m*_{i} = (4×8 ×10^{−6} ×900) = 288×10^{−4} kg

Latent heat of fusion of ice, *L*_{i} = 3.4 × 10^{5} J kg^{−1}

Density of the drink = 1000 kg m^{−3}

Volume of the drink = 200 ml

Mass of the drink = (200×10^{−6})×1000 kg

Let us first check the heat released when temperature of 200 ml changes from 10^{o}C to 0^{o}C.*H*_{w} = (200×10^{−6})×1000×4200×(10−0) = 8400 J

Heat required to change four 8 cm^{3} ice cubes into water (*H*_{i}) = *m*_{i}*L*_{i} = (288×10^{−4})×(3.4×10^{5}) = 9792 J

Since the heat required for melting the four cubes of the ice is greater than the heat released by water ( *H*_{i} > *H*_{w} ), some ice will remain solid and there will be equilibrium between

ice and water. Thus, the thermal equilibrium will be attained at 0^{o} C.

(b)

Equilibrium temperature of the cube and the drink = 0°C

Let *M* be the mass of melted ice.

Heat released when temperature of 200 ml changes from 10^{o}C to 0^{o}C is given by*H*_{w} = (200×10^{−6})×1000×4200×(10−0) = 8400 J

Thus,*M*×(3.4×10^{5}) = 8400 J

Therefore,*M *= 0.0247 Kg = 25 g

#### Answer:

Given:

Specific heat of water ,*S* = 4200 J kg^{−1} °C^{−1}

Latent heat of vapourisation of water ,*L* = 2.27 × 10^{6} J kg^{−1}

Mass,* M =* 0.2 g = 0.0002* *kg

Let us first calculate the amount of energy required to decrease the temperature of 10 kg of water by 5°C.*U*_{1} = 10 × 4200 J/kg°C × 5°C*U*_{1} = 210,000 = 21 × 10^{4} J

Let the time in which the temperature is decreased by 5°C be *t.*

Energy required per second for evaporation of water (at the rate of 0.2 g/sec) is given by*U*_{2}* = ML**U*_{2} = (2 × 10^{−4} )× (2.27 × 10^{6}) = 454 J

Total energy required to decrease the temperature of the water = 454 ×* t*

= 21 × 10^{4} J

Now,*t* $=\frac{21\times {10}^{4}}{454}$ seconds

The time taken in minutes is given by*t*$=\frac{21\times {10}^{4}}{454\times 60}=7.7\mathrm{minute}s$

∴ The time required to decrease the temperature by 5°C is 7.7 minutes.

#### Page No 47:

#### Question 6:

Given:

Specific heat of water ,*S* = 4200 J kg^{−1} °C^{−1}

Latent heat of vapourisation of water ,*L* = 2.27 × 10^{6} J kg^{−1}

Mass,* M =* 0.2 g = 0.0002* *kg

Let us first calculate the amount of energy required to decrease the temperature of 10 kg of water by 5°C.*U*_{1} = 10 × 4200 J/kg°C × 5°C*U*_{1} = 210,000 = 21 × 10^{4} J

Let the time in which the temperature is decreased by 5°C be *t.*

Energy required per second for evaporation of water (at the rate of 0.2 g/sec) is given by*U*_{2}* = ML**U*_{2} = (2 × 10^{−4} )× (2.27 × 10^{6}) = 454 J

Total energy required to decrease the temperature of the water = 454 ×* t*

= 21 × 10^{4} J

Now,*t* $=\frac{21\times {10}^{4}}{454}$ seconds

The time taken in minutes is given by*t*$=\frac{21\times {10}^{4}}{454\times 60}=7.7\mathrm{minute}s$

∴ The time required to decrease the temperature by 5°C is 7.7 minutes.

#### Answer:

Given:

Density of the iron cube = 8000 kg m^{$-$3}

Density of the ice cube = 900 kg m^{$-$3}

Specific heat capacity, *S *= 470 J kg^{−1} K^{−1}

Latent heat of fusion of ice, *L* = 3.36 × 10^{5} J kg^{−1}

Let the volume of the cube be *V*.

Volume of water displaced = *V*

Mass of cube, *m* = 8000 *V *kg

Mass of the ice melted, *M *= 900 *V*

Let the initial temperature of the iron cube be *T* K. Then,

Heat gained by the ice = Heat lost by the iron cube* m* × *S *× (*T* − 273) = *M × L*

⇒ 8000 *V* × 470 × (*T* − 273) = 900 *V*×( 3.36 × 10^{5})

⇒ 376 × 10^{4} × (*T* − 273) = 3024 × 10^{5}

$\Rightarrow T=\frac{30240+102648}{376}\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{132888}{376}\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=353.425\mathrm{K}\approx 353\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=353\mathrm{K}-273\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T\mathit{}=80\xb0\mathrm{C}$

#### Page No 47:

#### Question 7:

Given:

Density of the iron cube = 8000 kg m^{$-$3}

Density of the ice cube = 900 kg m^{$-$3}

Specific heat capacity, *S *= 470 J kg^{−1} K^{−1}

Latent heat of fusion of ice, *L* = 3.36 × 10^{5} J kg^{−1}

Let the volume of the cube be *V*.

Volume of water displaced = *V*

Mass of cube, *m* = 8000 *V *kg

Mass of the ice melted, *M *= 900 *V*

Let the initial temperature of the iron cube be *T* K. Then,

Heat gained by the ice = Heat lost by the iron cube* m* × *S *× (*T* − 273) = *M × L*

⇒ 8000 *V* × 470 × (*T* − 273) = 900 *V*×( 3.36 × 10^{5})

⇒ 376 × 10^{4} × (*T* − 273) = 3024 × 10^{5}

$\Rightarrow T=\frac{30240+102648}{376}\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{132888}{376}\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=353.425\mathrm{K}\approx 353\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=353\mathrm{K}-273\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T\mathit{}=80\xb0\mathrm{C}$

#### Answer:

Given:

Amount of ice at 0^{o}C = 1 kg

Amount of steam at 100^{o}C = 1 kg

Latent heat of fusion of ice = 3.36 × 10^{3} J kg^{−1}

Latent heat of vapourisation of water = 2.26 × 10^{6} J kg^{−1}

We can observe that the latent heat of fusion of ice (3.36 × 10^{5 }J kg^{−1}) is smaller that latent heat of vapouisation of water (2.26 × 10^{6} ). Therefore, ice will first change into water as less heat is required for this and there will be equilibrium between steam and water.

Heat absorbed by the ice when it changes into water (*Q*_{1}) = 1×(3.36 × 10^{5}) J

Heat absorbed by the water formed to change its temperature from 0^{o}C to 100^{o}C (*Q*_{2}) = 1 × 4200 × 100 = 4.2 × 10^{5} J

Total heat absorbed by the ice to raise the temperature to 100°C, *Q* = *Q*_{1}+*Q*_{2} = 3.36 × 10^{5} + 4.2 × 10^{5} = (3.36 + 4.2) × 10^{5} = 7.56 × 10^{5} J

The heat required to change ice into water at 100^{o}C is supplied by the steam. This heat will be released by the steam and will then change into water.

If all the steam gets converted into water, heat released by steam, *Q'* = 1 ×( 2.26 × 10^{6}) J = 2.26 × 10^{6} J

Amount of heat released is more than that required by the ice to get converted into water at 100^{o}C. Thus,

Extra heat = *Q* − *Q'*

= (2.26 − 0.756) × 10^{6}

= 1.506 × 10^{6}

Let the mass of steam that is condensed into water be *m*. Thus,*m*$=\frac{7.56\times {10}^{5}}{2.26\times {10}^{6}}$ = 0.335 kg = 335 gm

Total amount of water at 100°C = 1000 + 335 = 1335 g =1.335 g

Steam left = 1− 0.335 = 0.665 kg = 665 gm

#### Page No 47:

#### Question 8:

Given:

Amount of ice at 0^{o}C = 1 kg

Amount of steam at 100^{o}C = 1 kg

Latent heat of fusion of ice = 3.36 × 10^{3} J kg^{−1}

Latent heat of vapourisation of water = 2.26 × 10^{6} J kg^{−1}

We can observe that the latent heat of fusion of ice (3.36 × 10^{5 }J kg^{−1}) is smaller that latent heat of vapouisation of water (2.26 × 10^{6} ). Therefore, ice will first change into water as less heat is required for this and there will be equilibrium between steam and water.

Heat absorbed by the ice when it changes into water (*Q*_{1}) = 1×(3.36 × 10^{5}) J

Heat absorbed by the water formed to change its temperature from 0^{o}C to 100^{o}C (*Q*_{2}) = 1 × 4200 × 100 = 4.2 × 10^{5} J

Total heat absorbed by the ice to raise the temperature to 100°C, *Q* = *Q*_{1}+*Q*_{2} = 3.36 × 10^{5} + 4.2 × 10^{5} = (3.36 + 4.2) × 10^{5} = 7.56 × 10^{5} J

The heat required to change ice into water at 100^{o}C is supplied by the steam. This heat will be released by the steam and will then change into water.

If all the steam gets converted into water, heat released by steam, *Q'* = 1 ×( 2.26 × 10^{6}) J = 2.26 × 10^{6} J

Amount of heat released is more than that required by the ice to get converted into water at 100^{o}C. Thus,

Extra heat = *Q* − *Q'*

= (2.26 − 0.756) × 10^{6}

= 1.506 × 10^{6}

Let the mass of steam that is condensed into water be *m*. Thus,*m*$=\frac{7.56\times {10}^{5}}{2.26\times {10}^{6}}$ = 0.335 kg = 335 gm

Total amount of water at 100°C = 1000 + 335 = 1335 g =1.335 g

Steam left = 1− 0.335 = 0.665 kg = 665 gm

#### Answer:

Given:

Power rating of the immersion rod, *P* = 1000 W

Specific heat of water, *S* = 4200 J kg^{−1} K^{−1}

Mass of water, *M* = 20 kg

Change in temperature, Δ*T* = 25 °C

Total heat required to raise the temperature of 20 kg of water from 10°C to 35°C is given by*Q *= *M × S *× Δ*T**Q *= 20 × 4200 × 25*Q* = 20 × 4200 × 25 = 21 × 10^{5} J

Let the time taken to heat 20 kg of water from 10°C to 35°C be *t*. Only 80% of the immersion rod's heat is useful for heating water. Thus,

Energy of the immersion rod utilised for heating the water = *t* × (0.80) × 1000 J*t* × (0.80) × 1000 J = 21 × 10^{5} J

$t=\frac{21\times {10}^{5}}{800}=2625\mathrm{s}\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{2625}{60}=43.75\mathrm{min}\approx 44\mathrm{min}$

#### Page No 47:

#### Question 9:

Given:

Power rating of the immersion rod, *P* = 1000 W

Specific heat of water, *S* = 4200 J kg^{−1} K^{−1}

Mass of water, *M* = 20 kg

Change in temperature, Δ*T* = 25 °C

Total heat required to raise the temperature of 20 kg of water from 10°C to 35°C is given by*Q *= *M × S *× Δ*T**Q *= 20 × 4200 × 25*Q* = 20 × 4200 × 25 = 21 × 10^{5} J

Let the time taken to heat 20 kg of water from 10°C to 35°C be *t*. Only 80% of the immersion rod's heat is useful for heating water. Thus,

Energy of the immersion rod utilised for heating the water = *t* × (0.80) × 1000 J*t* × (0.80) × 1000 J = 21 × 10^{5} J

$t=\frac{21\times {10}^{5}}{800}=2625\mathrm{s}\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{2625}{60}=43.75\mathrm{min}\approx 44\mathrm{min}$

#### Answer:

Given:

Initial temperature of the water, *T*_{i}_{}= 20°C

Final temperature of the water (room temperature), *T*_{f}_{}= 5°C

Change in temeprature, Δ*T* = 20°C − 5°C = 15°C

Volume of water = 0.5 m^{3}

Density of water, d = 1000 kg/m^{3}

Mass of the water,* M* = (0.5 × 1000) kg = 500 kg

Heat liberated as the temperature of water changes from 20°C to 5°C is given by*Q* = *M×S*×Δ*T**Q *= (500 × 4200 × 15) J*Q *= (500 × 4200 × 15) J*Q *= (75 × 420 × 1000) J*Q *= 31500 × 1000 J = 315×10^{5} J

Let the height to which the mass is lifted be *h.*

The energy required to lift the block = *mgh* = 10 × 10 × *h* = 100 *h*

Acording to the question,*Q = mgh*

$\Rightarrow $100 *h* = 315×10^{5} J

$\Rightarrow $*h *= 315×10^{3}*m* = 315 km

#### Page No 47:

#### Question 10:

Given:

Initial temperature of the water, *T*_{i}_{}= 20°C

Final temperature of the water (room temperature), *T*_{f}_{}= 5°C

Change in temeprature, Δ*T* = 20°C − 5°C = 15°C

Volume of water = 0.5 m^{3}

Density of water, d = 1000 kg/m^{3}

Mass of the water,* M* = (0.5 × 1000) kg = 500 kg

Heat liberated as the temperature of water changes from 20°C to 5°C is given by*Q* = *M×S*×Δ*T**Q *= (500 × 4200 × 15) J*Q *= (500 × 4200 × 15) J*Q *= (75 × 420 × 1000) J*Q *= 31500 × 1000 J = 315×10^{5} J

Let the height to which the mass is lifted be *h.*

The energy required to lift the block = *mgh* = 10 × 10 × *h* = 100 *h*

Acording to the question,*Q = mgh*

$\Rightarrow $100 *h* = 315×10^{5} J

$\Rightarrow $*h *= 315×10^{3}*m* = 315 km

#### Answer:

Given:

Mass of bullet,* m *= 20g = 0.02 kg

Initial velocity of the bullet,* u* = 40 m/s

Final velocity of the bullet = 0 m/s

Initial kinetic energy of the bullet = $\frac{1}{2}m{u}^{2}=\frac{1}{2}\times 0.02\times 40\times 40=16\mathrm{J}$

Final kinetic energy of the bullet = 0

Change in energy of the bullet = 16 J

It is given that the bullet enters the block and stops inside it. The total change in its kinetic energy is responsible for the change in the internal energy of the block.

∴ Change in internal energy of the block = Change in energy of the bullet = 16 J

#### Page No 47:

#### Question 11:

Given:

Mass of bullet,* m *= 20g = 0.02 kg

Initial velocity of the bullet,* u* = 40 m/s

Final velocity of the bullet = 0 m/s

Initial kinetic energy of the bullet = $\frac{1}{2}m{u}^{2}=\frac{1}{2}\times 0.02\times 40\times 40=16\mathrm{J}$

Final kinetic energy of the bullet = 0

Change in energy of the bullet = 16 J

It is given that the bullet enters the block and stops inside it. The total change in its kinetic energy is responsible for the change in the internal energy of the block.

∴ Change in internal energy of the block = Change in energy of the bullet = 16 J

#### Answer:

Given:

Mass of the man, *m *= 50 kg

Speed of the man, *v *= 18 km/h = $18\times \frac{5}{18}=5\mathrm{m}/\mathrm{s}$

Kinetic energy of the man is given by*K*$=\frac{1}{2}m{\mathrm{V}}^{2}$

$K=\left(\frac{1}{2}\right)50\times {5}^{2}\phantom{\rule{0ex}{0ex}}K=25\times 25=625\mathrm{J}$

Specific heat of the water, *s *= 4200 J/Kg-K

Let the mass of the water heated be *M.*

The amount of heat required to raise the temperature of water from 20°C to 30°C is given by*Q* = *ms*Δ*T* = *M* × 4200 × (30 − 20)*Q* = 42000 *M*

According to the question,*Q = K*

42000 *M* = 625

$\Rightarrow M=\frac{625}{42}\times {10}^{-3}\phantom{\rule{0ex}{0ex}}=14.88\times {10}^{-3}\phantom{\rule{0ex}{0ex}}=15\mathrm{g}$

#### Page No 47:

#### Question 12:

Given:

Mass of the man, *m *= 50 kg

Speed of the man, *v *= 18 km/h = $18\times \frac{5}{18}=5\mathrm{m}/\mathrm{s}$

Kinetic energy of the man is given by*K*$=\frac{1}{2}m{\mathrm{V}}^{2}$

$K=\left(\frac{1}{2}\right)50\times {5}^{2}\phantom{\rule{0ex}{0ex}}K=25\times 25=625\mathrm{J}$

Specific heat of the water, *s *= 4200 J/Kg-K

Let the mass of the water heated be *M.*

The amount of heat required to raise the temperature of water from 20°C to 30°C is given by*Q* = *ms*Δ*T* = *M* × 4200 × (30 − 20)*Q* = 42000 *M*

According to the question,*Q = K*

42000 *M* = 625

$\Rightarrow M=\frac{625}{42}\times {10}^{-3}\phantom{\rule{0ex}{0ex}}=14.88\times {10}^{-3}\phantom{\rule{0ex}{0ex}}=15\mathrm{g}$

#### Answer:

Given:

Mass of the brick,* m* = 4 kg

Total vertical distance travelled by the brick, *h* = 3 m

Percentage of gravitational potential energy converted to thermal energy = 80

Total change in potential energy of the brick =* mgh* = 4 × 10 × 3 = 120 J

$\mathrm{Thermal}\mathrm{Energy}=120\times \frac{80}{100}=96\mathrm{J}$

Thermal energy in calories is given by

$U=\frac{96}{4.2}=22.857\mathrm{cal}\approx 23\mathrm{cal}$

#### Page No 47:

#### Question 13:

Given:

Mass of the brick,* m* = 4 kg

Total vertical distance travelled by the brick, *h* = 3 m

Percentage of gravitational potential energy converted to thermal energy = 80

Total change in potential energy of the brick =* mgh* = 4 × 10 × 3 = 120 J

$\mathrm{Thermal}\mathrm{Energy}=120\times \frac{80}{100}=96\mathrm{J}$

Thermal energy in calories is given by

$U=\frac{96}{4.2}=22.857\mathrm{cal}\approx 23\mathrm{cal}$

#### Answer:

Given:

Mass of van, *m* = 1500 kg

Speed of van, *v* = 54 km/h

$=54\times \left(\frac{5}{18}\right)=15\mathrm{m}/\mathrm{s}$

Total kinetic energy of the van is given by

$K=\frac{1}{2}m{v}^{2}$

$K=\frac{1}{2}\times 1500\times {\left(15\right)}^{2}\phantom{\rule{0ex}{0ex}}K=750\times 225\phantom{\rule{0ex}{0ex}}K=168750\mathrm{J}\phantom{\rule{0ex}{0ex}}K=\frac{168750}{4.2}\phantom{\rule{0ex}{0ex}}K=40178.57\mathrm{cal}$

Loss in total energy of the van = 40178 cal

Loss in energy per second $=\frac{40178}{10}=4017.8$

$\approx $ 4000 cal./sec

∴ Average rate of production of thermal energy ≈ 4000 cal/sec

#### Page No 47:

#### Question 14:

Given:

Mass of van, *m* = 1500 kg

Speed of van, *v* = 54 km/h

$=54\times \left(\frac{5}{18}\right)=15\mathrm{m}/\mathrm{s}$

Total kinetic energy of the van is given by

$K=\frac{1}{2}m{v}^{2}$

$K=\frac{1}{2}\times 1500\times {\left(15\right)}^{2}\phantom{\rule{0ex}{0ex}}K=750\times 225\phantom{\rule{0ex}{0ex}}K=168750\mathrm{J}\phantom{\rule{0ex}{0ex}}K=\frac{168750}{4.2}\phantom{\rule{0ex}{0ex}}K=40178.57\mathrm{cal}$

Loss in total energy of the van = 40178 cal

Loss in energy per second $=\frac{40178}{10}=4017.8$

$\approx $ 4000 cal./sec

∴ Average rate of production of thermal energy ≈ 4000 cal/sec

#### Answer:

Given:

Mass of the block = 100 g = 0.1 kg

Initial speed of the block = 10 m/s

Final speed of the block = 5 m/s

Initial kinetic energy of the block = $\frac{1}{2}\times 0.1\times {10}^{2}=5\mathrm{J}$

Final kinetic energy of the block = $\frac{1}{2}\times 0.1\times {5}^{2}=1.25\mathrm{J}$

Change in kinetic energy of the block = 5 $-$ 1.25 = 3.75 J

Thermal energy developed is equal to the change in kinetic energy of the block. Thus,

Thermal energy developed in the process = 3.75 J

#### Page No 47:

#### Question 15:

Given:

Mass of the block = 100 g = 0.1 kg

Initial speed of the block = 10 m/s

Final speed of the block = 5 m/s

Initial kinetic energy of the block = $\frac{1}{2}\times 0.1\times {10}^{2}=5\mathrm{J}$

Final kinetic energy of the block = $\frac{1}{2}\times 0.1\times {5}^{2}=1.25\mathrm{J}$

Change in kinetic energy of the block = 5 $-$ 1.25 = 3.75 J

Thermal energy developed is equal to the change in kinetic energy of the block. Thus,

Thermal energy developed in the process = 3.75 J

#### Answer:

Given:

Mass of the first block, *m*_{1} = 10 kg

Mass of the second block, *m*_{2} = 20 kg

Initial velocity of the first block, *u*_{1} = 10 m/s

Initial velocity of the second block, *u*_{2} = 20 m/s

Let the velocity of the blocks after collision be *v*.

Applying conservation of momentum, we get

* **m*_{2}*u*_{2} − *m*_{1}*u*_{1} = (*m*_{1} + *m*_{2})*v*

⇒ 20 × 20 − 10 × 10 = (10 + 20)*v*

⇒ 400 − 100 = 30 *v*

⇒ 300 = 30 *v*

⇒ *v* = 10 m/s

Initial kinetic energy is given by

${K}_{\mathrm{i}}=\frac{1}{2}{m}_{1}{u}_{1}^{2}+\frac{1}{2}{m}_{2}{u}_{2}^{2}$

${K}_{\mathrm{i}}=\frac{1}{2}\times 10\times {\left(10\right)}^{2}+\frac{1}{2}\times 20\times {\left(20\right)}^{2}\phantom{\rule{0ex}{0ex}}{K}_{\mathrm{i}}=500+4000=4500$

Final kinetic energy is given by

${K}_{\mathrm{f}}\mathit{=}\frac{1}{2}\left({m}_{1}\mathit{+}{m}_{2}\right){v}^{2}$

${K}_{\mathrm{f}}=\frac{1}{2}\left(10+20\right){\left(10\right)}^{2}\phantom{\rule{0ex}{0ex}}{K}_{\mathrm{f}}=\left(\frac{30}{2}\right)\times 100=1500$

∴ Total change in KE = 4500 J − 1500 J = 3000 J

Thermal energy developed in the process = 3000 J

#### Page No 47:

#### Question 16:

Given:

Mass of the first block, *m*_{1} = 10 kg

Mass of the second block, *m*_{2} = 20 kg

Initial velocity of the first block, *u*_{1} = 10 m/s

Initial velocity of the second block, *u*_{2} = 20 m/s

Let the velocity of the blocks after collision be *v*.

Applying conservation of momentum, we get

* **m*_{2}*u*_{2} − *m*_{1}*u*_{1} = (*m*_{1} + *m*_{2})*v*

⇒ 20 × 20 − 10 × 10 = (10 + 20)*v*

⇒ 400 − 100 = 30 *v*

⇒ 300 = 30 *v*

⇒ *v* = 10 m/s

Initial kinetic energy is given by

${K}_{\mathrm{i}}=\frac{1}{2}{m}_{1}{u}_{1}^{2}+\frac{1}{2}{m}_{2}{u}_{2}^{2}$

${K}_{\mathrm{i}}=\frac{1}{2}\times 10\times {\left(10\right)}^{2}+\frac{1}{2}\times 20\times {\left(20\right)}^{2}\phantom{\rule{0ex}{0ex}}{K}_{\mathrm{i}}=500+4000=4500$

Final kinetic energy is given by

${K}_{\mathrm{f}}\mathit{=}\frac{1}{2}\left({m}_{1}\mathit{+}{m}_{2}\right){v}^{2}$

${K}_{\mathrm{f}}=\frac{1}{2}\left(10+20\right){\left(10\right)}^{2}\phantom{\rule{0ex}{0ex}}{K}_{\mathrm{f}}=\left(\frac{30}{2}\right)\times 100=1500$

∴ Total change in KE = 4500 J − 1500 J = 3000 J

Thermal energy developed in the process = 3000 J

#### Answer:

Height of the floor from which ball is dropped, *h*_{1} = 2.0 m

Height to which the ball rises after collision, *h*_{2} = 1.5 m

Let the mass of ball be *m* kg.

Let the speed of the ball when it falls from *h*_{1} and *h*_{2} be *v*_{1}*and **v*_{2}, respectively.

${v}_{1}=\sqrt{2g{h}_{1}}=\sqrt{2\times 10\times 2}=\sqrt{40}\mathrm{m}/\mathrm{s}\phantom{\rule{0ex}{0ex}}{v}_{2}=\sqrt{2g{h}_{2}}=\sqrt{2\times 10\times 1.5}=\sqrt{30}\mathrm{m}/\mathrm{s}$

Change in kinetic energy is given by

$\u2206K=\frac{1}{2}\times \mathrm{m}\times 40-\left(\frac{1}{2}\mathrm{m}\right)\times 30=\left(\frac{10}{2}\right)\mathrm{m}\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206K=5\mathrm{m}$

If the position of the ball is considered just before hitting the ground and after its first collision, then 40% of the change in its KE will give the change in thermal energy of the ball. At these positions, the PE of the ball is same. Thus,

Loss in PE = 0

The change in kinetic energy is utilised in increasing the temperature of the ball.

Let the change in temperature be Δ*T**.* Then,

$\left(\frac{40}{100}\right)\times \u2206K=m\times 800\times \u2206T\phantom{\rule{0ex}{0ex}}\left(\frac{40}{100}\right)\times \frac{10}{2}m=m\times 800\times \u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206T=\frac{1}{400}=0.0025\phantom{\rule{0ex}{0ex}}=2.5\times {10}^{-3}\xb0\mathrm{C}$

#### Page No 47:

#### Question 17:

Height of the floor from which ball is dropped, *h*_{1} = 2.0 m

Height to which the ball rises after collision, *h*_{2} = 1.5 m

Let the mass of ball be *m* kg.

Let the speed of the ball when it falls from *h*_{1} and *h*_{2} be *v*_{1}*and **v*_{2}, respectively.

${v}_{1}=\sqrt{2g{h}_{1}}=\sqrt{2\times 10\times 2}=\sqrt{40}\mathrm{m}/\mathrm{s}\phantom{\rule{0ex}{0ex}}{v}_{2}=\sqrt{2g{h}_{2}}=\sqrt{2\times 10\times 1.5}=\sqrt{30}\mathrm{m}/\mathrm{s}$

Change in kinetic energy is given by

$\u2206K=\frac{1}{2}\times \mathrm{m}\times 40-\left(\frac{1}{2}\mathrm{m}\right)\times 30=\left(\frac{10}{2}\right)\mathrm{m}\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206K=5\mathrm{m}$

If the position of the ball is considered just before hitting the ground and after its first collision, then 40% of the change in its KE will give the change in thermal energy of the ball. At these positions, the PE of the ball is same. Thus,

Loss in PE = 0

The change in kinetic energy is utilised in increasing the temperature of the ball.

Let the change in temperature be Δ*T**.* Then,

$\left(\frac{40}{100}\right)\times \u2206K=m\times 800\times \u2206T\phantom{\rule{0ex}{0ex}}\left(\frac{40}{100}\right)\times \frac{10}{2}m=m\times 800\times \u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206T=\frac{1}{400}=0.0025\phantom{\rule{0ex}{0ex}}=2.5\times {10}^{-3}\xb0\mathrm{C}$

#### Answer:

Mass of copper cube, *m* = 200 g = 0.2 kg

Length through which the block has slided, *l* = 60 cm = 0.6 m

Since the block is moving with constant velocity, the net force on it is zero. Thus,

Force of friction, *f* = mg

Also, since the object is moving with a constant velocity, change in its K.E will be zero.As the object slides down, its PE decreases at the cost of increase in thermal energy of copper.

The loss in mechanical energy of the copper block = Work done by the frictional force on the copper block to a distanceof 60 cm*W* = *mg* *l* sin *θ**W* = 0.2 × 10 × 0.6 sin 37°*W* $=1.2\times \left(\frac{3}{5}\right)=0.72$

Let the change in temperature of the block be âˆ†*T*.

Thermal energy gained by block = *ms* âˆ†*T* = 0.2 × 420×âˆ†*T* = 84âˆ†*T*

But 84âˆ†*T* = 0.72

$\Rightarrow \u2206T=\frac{0.72}{84}=0.00857\phantom{\rule{0ex}{0ex}}\u2206T=0.0086=8.6\times {10}^{-3}\xb0\mathrm{C}$

#### Page No 47:

#### Question 18:

Mass of copper cube, *m* = 200 g = 0.2 kg

Length through which the block has slided, *l* = 60 cm = 0.6 m

Since the block is moving with constant velocity, the net force on it is zero. Thus,

Force of friction, *f* = mg

Also, since the object is moving with a constant velocity, change in its K.E will be zero.As the object slides down, its PE decreases at the cost of increase in thermal energy of copper.

The loss in mechanical energy of the copper block = Work done by the frictional force on the copper block to a distanceof 60 cm*W* = *mg* *l* sin *θ**W* = 0.2 × 10 × 0.6 sin 37°*W* $=1.2\times \left(\frac{3}{5}\right)=0.72$

Let the change in temperature of the block be âˆ†*T*.

Thermal energy gained by block = *ms* âˆ†*T* = 0.2 × 420×âˆ†*T* = 84âˆ†*T*

But 84âˆ†*T* = 0.72

$\Rightarrow \u2206T=\frac{0.72}{84}=0.00857\phantom{\rule{0ex}{0ex}}\u2206T=0.0086=8.6\times {10}^{-3}\xb0\mathrm{C}$

#### Answer:

Given:

Density of metal block, *d* = 600 kg m^{−3}

Mass of metal block, *m* = 1.2 kg

Spring constant of the spring,* k* = 200 N m^{−1}

Volume of the block, *V*$=\frac{1.2}{6000}=2\times {10}^{-4}{\mathrm{m}}^{3}$

When the mass is dipped in water, it experiences a buoyant force and in the spring there is potential energy stored in it.

If the net force on the block is zero before breaking of the support of the spring, then*kx* + *V*ρ*g* = *mg*

200*x* + (2 × 10^{−4})× (1000) × (10) = 12

$\Rightarrow x=\frac{\left(12-2\right)}{200}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{10}{200}=0.05\mathrm{m}$

The mechanical energy of the block is transferred to both block and water. Let the rise in temperature of the block and the water be Δ*T*.

Applying conservation of energy, we get

$\frac{1}{2}k{x}^{2}+mgh-V\mathrm{\rho}gh={m}_{1}{s}_{1}\u2206T+{m}_{2}{s}_{2}\u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}\times 200\times 0.0025+1.2\times 10\times \left(\frac{40}{100}\right)-2\times {10}^{-4}\times 1000\times 10\times \left(\frac{40}{100}\right)\phantom{\rule{0ex}{0ex}}=\left(\frac{260}{1000}\right)\times 4200\times \u2206T+1.2\times 250\times \u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow 0.25+4.8-0.8=1092\u2206T+300\u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow 1392\u2206T=4.25\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206T=\frac{4.25}{1392}=0.0030531\phantom{\rule{0ex}{0ex}}\u2206T=3\times {10}^{-3}\xb0\mathrm{C}$

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