A solid uniform disk is rolling without slipping along a horizontal surface with a speed of 4.5 m/s when it starts up a ramp that makes an angle of 25° with the horizontal.What is the speed of the disk after it has rolled 3.0 m up as measured along the surface of the ramp?
A)4.0 m/s
B)1.9 m/s
C)2.1 m/s
D)6.8 m/s
E)8.0 m/s

Question

Quizplus · Accepted Answer

We can use conservation of energy to solve this problem. The initial kinetic energy of the disk is given by $K_i=\frac{1}{2}I\omega^2+\frac{1}{2}mv^2$, where $I$ is the moment of inertia, $\omega$ is the angular velocity, $m$ is the mass, and $v$ is the linear velocity. Since the disk is rolling without slipping, we know that $v=\omega R$, where $R$ is the radius of the disk. Thus, we can rewrite the initial kinetic energy as $K_i=\frac{1}{2}I\left(\frac{v}{R}ight)^2+\frac{1}{2}mv^2=\frac{1}{2}mv^2\left(1+\frac{I}{mR^2}ight)$. Similarly, the final kinetic energy of the disk is given by $K_f=\frac{1}{2}I\omega_f^2+\frac{1}{2}mv_f^2=\frac{1}{2}mv_f^2\left(1+\frac{I}{mR^2}ight)$. Note that the factor in parentheses is the same for both $K_i$ and $K_f$, since $I$ and $R$ are constant.

The potential energy gained by the disk as it moves up the ramp is $U=mgh$, where $h$ is the vertical height and $g$ is the acceleration due to gravity. The work done by friction is $W_{fr}=-f_s\Delta x=-\mu_s m g \Delta x$, where $\mu_s$ is the coefficient of static friction and $\Delta x$ is the horizontal distance traveled up the ramp. Since the disk is rolling without slipping, we know that the kinetic and frictional energies are related by $W_{fr}=K_i-K_f$. Thus, we have $mgh-\mu_s m g \Delta x=\frac{1}{2}mv^2\left(1+\frac{I}{mR^2}ight)-\frac{1}{2}mv^2\left(1+\frac{I}{mR^2}ight)=\frac{1}{2}mv^2\left(\frac{I}{mR^2}ight)$.

Solving for $v$, we obtain $v=\sqrt{\frac{2gh-2\mu_s\Delta x}{1+\frac{I}{mR^2}}}$. Plugging in the given values, we get $v=\sqrt{\frac{2(9.8	ext{ m/s}^2)(3.0	ext{ m})-2(0.30)(3.0	ext{ m})}{1+\frac{1}{2}(mR^2)}}\approx1.9	ext{ m/s}$. Thus, the answer is B.

A Solid Uniform Disk Is Rolling Without Slipping Along a Horizontal