Question

Non-uniform cylindrical object.InFigure, a cylindrical object of massMand radius Rrolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance d = 0.506m from the end of the ramp. The initial height of the object is H = 0.90m ; the end of the ramp is at height h = 0.10m . The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form $\mathit{l}\mathbf{=}\mathit{\beta }\mathit{M}{\mathit{R}}^{\mathbf{2}}$but b is not 0.5as it is for a cylinder of uniform density. Determine $\mathit{\beta }$.

Short answer

Value of $\beta$ is 0.25

Step-by-step solution

Identification of given data

1. The horizontal distance d = 0.506m
2. Initial height of ramp H = 0.90m ,
3. New height of the ramp h = 0.10m

To understand the concept

This is based on the concept of energy conservation and projectile motion. Here, the body is made to freely along an inclined plane then, it gets projected from a height, h to drop at a distance from the ramp. Thus, the rolling of the cylindrical shell gives us the involvement of angular velocity to get the moment of inertia of the shell along its axis. Thus, the required term that is $\mathit{\beta }$.

Formulae:

The distance travelled by the body according to second law of kinematics,

$s=Vt+\frac{1}{2}g{t}^2$

where, V is the initial velocity of the body, g is the acceleration due to gravity, t is the time taken.

The transverse velocity of the body,

$V_{com}=R_{\omega }$

Where, R is the radius of the circular path, $\omega$is the angular velocity of the body.

The time of flight of the projected body,

$t=\sqrt{\frac{2h}{g}}$

where, h is the height of the flight

Determine the value of β 

By using energy conservation, we can write

$MgH=\frac{1}{2}l{\omega }^2+\frac{1}{2}m{v}_0^2+mgh$

As the cylinder would have potential energy only at height H. However, at new height h, part of the energy would be rotational kinetic energy, translational kinetic energy and potential energy.

Using, $v=r\omega$

$\begin{array}{rcl}Mg\left(H-h\right)& =& \frac{1}{2}l\left(\frac{v^2}{R^2}\right)+\frac{1}{2}m{v}_0^2\\ & =& \frac{1}{2}{v}_0^2\left(\frac{1}{R^2}+M\right)\\ \frac{2Mg\left(H-h\right)}{v_0^2}& =& \left(\frac{1}{R^2}+M\right)\\ \frac{2Mg\left(H-h\right)}{v_0^2}-M& =& \frac{1}{R^2}\\ M{R}^2\left(\frac{2g\left(H-h\right)}{v_0^2}-1\right)& =& l_{com}\\ l_{com}& =& M{R}^2\left[\frac{2g\left(H-h\right)}{v_0^2}-1\right]\end{array}$

In addition, we have

$l_{com}=\beta M{R}^2$

We can say that

$\beta =\begin{array}{rcl}& & \frac{2g\left(H-h\right)}{v_0^2}\end{array}\begin{array}{rcl}& & -\end{array}\begin{array}{rcl}& & 1\end{array}$

We know the vertical component of velocity is zero, we can say that V_0y = 0 and by using kinematic equation, we have

$\begin{array}{rcl}∆y& =& v_{i}t+\frac{1}{2}a{t}^2\\ h& =& \frac{1}{2}g{t}^2\\ t& =& \sqrt{\frac{2h}{g}}\end{array}$

Now,

$\begin{array}{rcl}s& =& v_{o}t+\frac{1}{2}g{t}^2\\ d& =& v_o\times \sqrt{\frac{2h}{g}}\\ v_0^2& =& \frac{g{d}^2}{2h}\end{array}$

After plugging the value of velocity in equation of $\beta$we get,

$\begin{array}{rcl}\beta & =& \frac{2g\left(H-h\right)}{v_0^2}-1\\ & =& \frac{2g\left(H-h\right)}{{\displaystyle \frac{g{d}^2}{2h}}}-1\\ & =& \frac{4h\left(H-h\right)}{d^2}-1\\ & =& \frac{4\left(0.10m\right)\left(0.80m\right)}{0.256m^2}-1\\ & =& 0.25\end{array}$