A solid sphere is rolling without slipping along a horizontal surface with a speed of 5.50 m/s when it starts up a ramp that makes an angle of 25.0° with the horizontal. What is the speed of the sphere after it has rolled 3.00 m up the ramp?
A) 4.01 m/s
B) 8.02 m/s
C) 1.91 m/s
D) 2.16 m/s
E) 3.53 m/s

Question

Quizplus · Accepted Answer

When a solid sphere rolls without slipping, its kinetic energy is given by the sum of its translational and rotational kinetic energies:

KE = 1/2 mv^2 + 1/2 Iω^2

where m is the mass of the sphere, v is its linear speed, I is its moment of inertia, and ω is its angular velocity.

In this case, because the sphere is rolling without slipping, we have

v = Rω

where R is the radius of the sphere. We can use this relationship to eliminate ω in favor of v in the equation above:

KE = 1/2 mv^2 + 1/2 (2/5 mR^2)(v/R)^2
   = 1/2 mv^2 + 1/5 mv^2
   = 7/10 mv^2

Note that we've used the fact that for a solid sphere, I = 2/5 mR^2.

Now, as the sphere rolls up the ramp, it gains gravitational potential energy:

PE = mgh

where h is the height it has climbed. Conservation of energy tells us that the increase in the sphere's potential energy must come at the expense of its kinetic energy, so we have

mgh = KEi - KEf

where KEi is the sphere's initial kinetic energy (when it is rolling without slipping along the horizontal surface) and KEf is its final kinetic energy (when it has rolled up the ramp).

At the start of the ramp, the sphere has only translational kinetic energy, so

KEi = 1/2 mv^2

At the top of the ramp, the sphere has both translational and rotational kinetic energy, but because it is rolling without slipping, we have

v = ωR

and

KEf = 1/2 mv^2 + 1/2 Iω^2
    = 1/2 mv^2 + 1/2 (2/5 mR^2)(v/R)^2
    = 1/2 mv^2 + 1/5 mv^2
    = 7/10 mv^2

(where we've used the same calculations as before to find the rotational kinetic energy).

Putting it all together:

mgh = 1/2 mv^2 - 7/10 mv^2

Solving for v:

v = sqrt[(2gh)/7]
  = sqrt[(2 * 9.81 m/s^2 * 3.00 m)/7]
  ≈ 3.53 m/s

So the answer is E.

A Solid Sphere Is Rolling Without Slipping Along a Horizontal