1. A solid sphere of mass *M* and radius *R* rolls without slipping down a ramp inclined at an angle *θ* to the horizontal. The sphere starts from rest at a vertical height *h* above the bottom of the ramp. The moment of inertia of a solid sphere about its centre is *I* = (2/5)*MR*2.
(a) Show that the total kinetic energy *E*K of the sphere when it reaches the bottom of the ramp is given by *E*K = (7/10)*Mv*2, where *v* is the translational speed of the sphere’s centre of mass.
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(b) Using conservation of energy, determine the translational speed *v* of the sphere at the bottom of the ramp in terms of *g* and *h*.
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(c) Another object, a hollow cylinder of the same mass *M* and radius *R*, is released from rest from the same height *h* on the same ramp. The moment of inertia of a hollow cylinder is *I* = *MR*2. Without further calculation, state whether the hollow cylinder will reach the bottom of the ramp with a translational speed greater than, less than, or equal to the speed of the solid sphere. Explain your reasoning.
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[VISUALIZATION: Diagram showing a solid sphere at the top of an inclined ramp of angle theta and vertical height h. Arrow indicating rolling down the ramp.]
Part a
4 minutes
3 marks
Question
Show that the total kinetic energy *E*K of the sphere when it reaches the bottom of the ramp is given by *E*K = (7/10)*Mv*2, where *v* is the translational speed of the sphere’s centre of mass.
Hint Hint
Hint
Start by writing the expression for total kinetic energy as the sum of translational and rotational kinetic energy. Substitute the given moment of inertia and the condition for rolling without slipping (\(v = R\omega\)).
Using conservation of energy, determine the translational speed *v* of the sphere at the bottom of the ramp in terms of *g* and *h*.
Hint Hint
Hint
Equate the initial gravitational potential energy at height *h* to the total kinetic energy at the bottom of the ramp (found in part a).
Answer Answer
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Explanation Explanation
Answer Answer
Complete Answer
\\(v = \\sqrt{\\frac{10gh}{7}}\\)
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Answer Explanation Answer Explanation
Part c
5 minutes
3 marks
Question
Another object, a hollow cylinder of the same mass *M* and radius *R*, is released from rest from the same height *h* on the same ramp. The moment of inertia of a hollow cylinder is *I* = *MR*2. Without further calculation, state whether the hollow cylinder will reach the bottom of the ramp with a translational speed greater than, less than, or equal to the speed of the solid sphere. Explain your reasoning.
Hint Hint
Hint
Compare the moment of inertia of the hollow cylinder to that of the solid sphere. Consider how this difference affects the distribution of the initial potential energy into translational and rotational kinetic energy.
Answer Answer
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Explanation Explanation
Answer Answer
Complete Answer
Less than. The hollow cylinder has a larger moment of inertia (\\(I_{cyl} = MR^2 > I_{sph} = \\frac{2}{5}MR^2\\)). For the same initial potential energy (\\(Mgh\\)), a larger fraction is converted into rotational kinetic energy (\\(\\frac{1}{2}I\\omega^2\\)) and a smaller fraction into translational kinetic energy (\\(\\frac{1}{2}Mv^2\\)), resulting in a lower translational speed \\(v\\).
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