Question

The diameter of ball $P$ is three times that of another ball $Q$ of the same material. $P$ and $Q$ are heated to same temperature and allowed to cool up to the room temperature. The relation between their rates of cooling will be :

• A. $R_P=R_Q/3$
• B. $R_P=3R_Q$
• C. $R_P=9R_Q$
• D. $R_P=R_Q$

Answer 1

The correct option is C $R_P=9R_Q$

Newton's law of cooling states that:
$\frac{dT}{dt}=-bA(T-T_0)=R$; where $T_{0}isroomtemperature$
R is the rate of cooling.
$A=4\pi R^2=\pi D^2$
Since both the balls are heated to the same temperature and are of the same material.
$R_P=-b\pi D_P^2(T-T_0)$
$R_Q=-b\pi D_Q^2(T-T_0)$
Given that the diameter of P is three times that of Q.
$D_P=3D$
$D_Q=D$
$R_P=-9b\pi D^2(T-T_0)$
$R_Q=-b\pi D^2(T-T_0)$
i.e. $R_P=9R_Q$