Spheres $P$ and $Q$ are uniformly constructed from the same material which is a good conductor of heat and the radius of $Q$ is thrice the radius of $P$ . The rate of fall of temperature of $P$ is $x$ times that of $Q$ . When both are at the same surface temperature, the value of $x$ is :

Question

Verified by Toppr

Answer 1

From Newton's law of cooling,
$\frac{d T}{d t} = - \frac{4 σ e A T _{0}^{3}}{m s} (T - T_{0}) = - \frac{4 σ e ( 4 π r ^{2} ) T _{0}^{3}}{( ρ 4 / 3 π r ^{3} ) s} (T - T_{0})$
Thus, $\frac{d T}{d t} \propto \frac{1}{r}$

$\frac{( \frac{d T}{d t} ) _{P}}{( \frac{d T}{d t} ) _{Q}} = \frac{r _{Q}}{r _{P}} = \frac{3 r _{P}}{r _{P}} = 3$
$∴ (\frac{d T}{d t})_{P} = 3 \times (\frac{d T}{d t})_{Q}$

Answer 2

From Newton's law of cooling,

\frac{d T}{d t} = - \frac{4 σ e A T _{0}^{3}}{m s} (T - T_{0}) = - \frac{4 σ e ( 4 π r ^{2} ) T _{0}^{3}}{( ρ 4 / 3 π r ^{3} ) s} (T - T_{0})

Thus,

\frac{d T}{d t} \propto \frac{1}{r}

\frac{( \frac{d T}{d t} ) _{P}}{( \frac{d T}{d t} ) _{Q}} = \frac{r _{Q}}{r _{P}} = \frac{3 r _{P}}{r _{P}} = 3

∴ (\frac{d T}{d t})_{P} = 3 \times (\frac{d T}{d t})_{Q}

Spheres $P$ and $Q$ are uniformly constructed from the same material which is a good conductor of heat and the radius of $Q$ is thrice the radius of $P$ . The rate of fall of temperature of $P$ is $x$ times that of $Q$ . When both are at the same surface temperature, the value of $x$ is :

$\frac{1}{4}$

$\frac{1}{3}$

$3$

$4$

Solution

Patterns of problems

Was this answer helpful?

Similar questions

A spherical black body of radius $r$ radiates power $P$ , and its rate of cooling is $R$ .

A sphere of radius $R$ , density $D$ and specific heat $S$ is heated to temperature $θ$ and surrounding is at temperature $θ_{0}$ . Its rate of fall of temperature will be proportional to

If the rates of cooling of two bodies are same then for which body will the rate of fall of temperature be more? For the body whose thermal capacity is

Two spheres of equal masses , one made up of lead and the other of copper have identical surface finish. If these two are heated upto the same temperature $θ$ and allowed to cool in the surrounding temperature $θ_{0}$ , then the total amount of heat loss during the same time will be

A solid sphere and a hollow sphere of same material and same radius are kept at same temperatures in atmosphere. Rate of cooling of hollow sphere will be more.

If all other conditions are same, then rate of cooling is inversely proportional to mass of body.

More From Chapter

Thermal Properties of Matter

Shortcuts & Tips

Common Misconceptions

Mindmap

Diagram set

Problem solving tips

Memorization tricks

Practice more questions

4 Qs

1 Qs

2 Qs

3 Qs

168 Qs

596 Qs

Spheres P and Q are uniformly constructed from the same material which is a good conductor of heat and the radius of Q is thrice the radius of P. The rate of fall of temperature of P is x times that of Q. When both are at the same surface temperature, the value of x is :

41​

31​

3

4

From Newton's law of cooling,dtdT​=−ms4σeAT03​​(T−T0​)=−(ρ4/3πr3)s4σe(4πr2)T03​​(T−T0​) Thus, dtdT​∝r1​ (dtdT​)Q​(dtdT​)P​​=rP​rQ​​=rP​3rP​​=3 ∴(dtdT​)P​=3×(dtdT​)Q​

Patterns of problems

Was this answer helpful?

Similar questions

A spherical black body of radius r radiates power P, and its rate of cooling is R.

A sphere of radius R, density D and specific heat S is heated to temperature θ and surrounding is at temperature θ0​. Its rate of fall of temperature will be proportional to

If the rates of cooling of two bodies are same then for which body will the rate of fall of temperature be more? For the body whose thermal capacity is

Two spheres of equal masses , one made up of lead and the other of copper have identical surface finish. If these two are heated upto the same temperature θ and allowed to cool in the surrounding temperature θ0​, then the total amount of heat loss during the same time will be

A solid sphere and a hollow sphere of same material and same radius are kept at same temperatures in atmosphere. Rate of cooling of hollow sphere will be more.

If all other conditions are same, then rate of cooling is inversely proportional to mass of body.

More From Chapter

Thermal Properties of Matter

Shortcuts & Tips

Common Misconceptions

Mindmap

Diagram set

Problem solving tips

Memorization tricks

Practice more questions

4 Qs

1 Qs

2 Qs

3 Qs

168 Qs

596 Qs

Spheres $P$ and $Q$ are uniformly constructed from the same material which is a good conductor of heat and the radius of $Q$ is thrice the radius of $P$ . The rate of fall of temperature of $P$ is $x$ times that of $Q$ . When both are at the same surface temperature, the value of $x$ is :

$\frac{1}{4}$

$\frac{1}{3}$

$3$

$4$

A spherical black body of radius $r$ radiates power $P$ , and its rate of cooling is $R$ .

A sphere of radius $R$ , density $D$ and specific heat $S$ is heated to temperature $θ$ and surrounding is at temperature $θ_{0}$ . Its rate of fall of temperature will be proportional to

Two spheres of equal masses , one made up of lead and the other of copper have identical surface finish. If these two are heated upto the same temperature $θ$ and allowed to cool in the surrounding temperature $θ_{0}$ , then the total amount of heat loss during the same time will be