Question 433: Properties of Matter

Question

A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as $$T(t) = {T_0}\left( {1 + \beta {t^{{1 \over 4}}}} \right)$$, where $$\beta $$ is a constant with appropriate dimension while T
0
is a constant with dimension of temperature. The heat capacity of the metal is

Accepted Answer

Heat capacity, $${{dQ} \over {dt}} = H{{dT} \over {dt}}$$
Power of the rod, $$P = H.{T_0}.\beta .{1 \over 4}.{t^{ - 3/4}}$$
$${{4P} \over {{T_0}.\beta }} = {t^{ - 3/4}}.H$$
$$ \Rightarrow $$ $$H = \left( {{{4P} \over {{T_0}\beta }}} \right){t^{3/4}}$$ .... (i)
Now, $$T - {T_0} = {T_0}\beta {t^{1/4}}$$
So, $${t^{3/4}} = {\left( {{{T - {T_0}} \over {{T_0}\beta }}} \right)^3}$$
Substituting this value of t
3/4
in equation (i) we get,
$$H = {{4P{{(T - {T_0})}^3}} \over {T_0^4{\beta ^4}}}$$

Answer

$${{4P{{(T(t) - {T_0})}^4}} \over {{\beta ^4}T_0^5}}$$

Answer

$${{4P{{(T(t) - {T_0})}^3}} \over {{\beta ^4}T_0^4}}$$

Answer

$${{4P(T(t) - {T_0})} \over {{\beta ^4}T_0^2}}$$

Answer

$${{4P{{(T(t) - {T_0})}^2}} \over {{\beta ^4}T_0^3}}$$

Properties of Matter

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