What Is the Effect of Friction on the Height of a Sphere Rolling Down a Ramp?

[Lone Wolf]

Homework Statement

A small sphere of radius r = 5 cm, with mass = 50 g, is dropped from rest from the top of a ramp with height = 0.73 m, as show in the figure. The moment of inertia of a sphere about its center is I = 2/5 MR². Consider g = 10 m/s².
a) Considering a sphere that rolls without slipping, find the speed of the sphere at point A (0.1 m above the base of the ramp).
b) Find the maximum height h, reached by the sphere.
c) Considering the sphere now only slips (does not roll), determine whether the maximum height will be greater or less than the previous case, without doing any calculations.

Relevant Equations

Conservation of mechanical energy: ΔEm = 0.
Kinetic energy for a rolling object: 1/2 * I * ω² + 1/2 * m * v² , where v is the velocity of the center of mass.
Potential energy = m * g * h, where h is the height of the center of mass.

Figure:

a)
The mechanical energy of the sphere is conserved because the weight is the only force which does work. My problem with this question is mostly because the original picture (which I tried to recreate here) is kind of ambiguous, as in I don't know if H already accounts for the radius of the sphere or not.
Initial potential energy: m * g * hi = 0.05 * 10 * 0.73 = 0.365 J.
Final potential energy: m * g * hf = 0.05 * 10 * (0.1) = 0.05 J.
Initial kinetic energy = 0 J
Final kinetic energy = 1/2 * I * ω² + 1/2 * m * v²; for an object that rolls without slipping, ω = v/R.
Kf = (7 * m * v²)/10 = 0.035 * v²
K(initial) + Ep(initial) = Ep(final) + K(final)
0.365 = 0.05 + 0.035v² ---> v = 3 m/s.
b)
Since the question does not mention any air resistance, the conservation of mechanical energy is still applicable. So the height of the sphere will be the same as it was initially (H = 0.73 m).
c) In order for an object to roll without slipping, there must be a friction force. So considering the ramp is not frictionless, as the sphere slides without rolling, it loses mechanical energy due to work done by the friction force, at point A the speed will be less than the one calculated for a), and therefore the sphere will reach a smaller height than 0.73 m.

I am not sure if the work I've done is correct. Any confirmations/corrections are appreciated.

 

[TSny]

a)
The mechanical energy of the sphere is conserved because the weight is the only force which does work. My problem with this question is mostly because the original picture (which I tried to recreate here) is kind of ambiguous, as in I don't know if H already accounts for the radius of the sphere or not.

Good point. From the way in which the problem is worded, I don't think that they intended for you to worry about this. But I would need to see the original figure to be sure.

Initial potential energy: m * g * hi = 0.05 * 10 * 0.73 = 0.365 J.
Final potential energy: m * g * hf = 0.05 * 10 * (0.1) = 0.05 J.
Initial kinetic energy = 0 J
Final kinetic energy = 1/2 * I * ω² + 1/2 * m * v²; for an object that rolls without slipping, ω = v/R.
Kf = (7 * m * v²)/10 = 0.035 * v²
K(initial) + Ep(initial) = Ep(final) + K(final)
0.365 = 0.05 + 0.035v² ---> v = 3 m/s.

OK. This looks good. (If you work out the problem in symbols and wait until the end to plug in numbers, you will see that the mass cancels and you will also avoid introducing inaccuracies due to round-off errors.)

b)
Since the question does not mention any air resistance, the conservation of mechanical energy is still applicable. So the height of the sphere will be the same as it was initially (H = 0.73 m).

This might not be correct. Hint: When the sphere reaches its maximum height, does the sphere have only gravitational potential energy, or does it still have some kinetic energy?

c) In order for an object to roll without slipping, there must be a friction force. So considering the ramp is not frictionless, as the sphere slides without rolling, it loses mechanical energy due to work done by the friction force, at point A the speed will be less than the one calculated for a), and therefore the sphere will reach a smaller height than 0.73 m.

I'm a little confused with your answer here. Part (a) is where friction is acting. In part (c), I think they want you to assume that the sphere slides without friction.

 

[TSny]

If the figure is equivalent to the one below, then you would need to include the radius of the sphere in determining the change in height of the sphere in going from the initial position to point A.