Question

For a body rolling along a level surface. Without slipping the translation and rotational kinetic energy are in the ratio. The body is
A. Hollow sphere
B. Solid cylinder
C. Ring
D. Solid sphere

Answer 1

To respond to the question, which is founded on the principle of rolling without slipping. Without slipping, rolling entails a combination of translation and rotation, with the point of contact being stationary at all times. We'll figure out what the object is by calculating the ratio of these two types of energy and adjusting the values accordingly.

Complete answer:
Let us consider that the moment of inertia is given as $I$ , mass of the body is given by $m$ , radius and velocity of the body is given as $R$ and $v$ respectively.
The kinetic energy of a rolling object is both translational and rotational.
Now, it is said that body is rolling without slipping, hence in this case velocity is given as;
$v = R\omega$
Therefore now, translational kinetic energy i.e. translational kinetic energy of a body is equal to one-half the product of its mass, m, and the square of its velocity, $v$ , or $\dfrac{1}{2}m{v^2}.\;$
And rotational kinetic energy is proportional to the square of the angular velocity magnitude and angular velocity. i.e. $\dfrac{1}{2}I{\omega ^2} = \dfrac{{I{v^2}}}{{2{R^2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\because \omega = \dfrac{{{v^2}}}{{{R^2}}}} \right]$
Now it is given to us in the question the ratio of both the energies i.e. $1:2$
Now by the help of that is ratio of energies we will determine our answer $\dfrac{{I{v^2}}}{{2{R^2}}}\,$
$\Rightarrow \dfrac{{TKE}}{{RKE}} = 2 \\\ \Rightarrow 2 = \dfrac{{\dfrac{1}{2}m{v^2}}}{{\dfrac{{I{v^2}}}{{2{R^2}}}}} \\\$
Therefore from here we will get the value of $I$ I,e,;
$\therefore I = \dfrac{1}{2}m{R^2}$
Hence, body is a solid cylinder as the moment of inertia of a solid cylinder about its centre is given by the formula; $I = \dfrac{1}{2}m{R^2}$
The correct option is: (B) Solid cylinder.

Note:
It’s worth mentioning that the total energy of a rolling item without slipping is conserved and constant during the motion in the absence of any non-conservative forces that would pull energy out of the system in the form of heat.