Suppose a solid sphere of mass m and radius R rolls without slipping down an inclined plane starting from rest. The angular velocity of the sphere at the bottom of the incline depends on

Question

Suppose a solid sphere of mass m and radius R rolls without slipping down an inclined plane starting from rest. The angular velocity of the sphere at the bottom of the incline depends on

BlackTom AI · Accepted Answer

Rolling without slipping links translational speed v and angular speed ω by v = ωR.

1) Relevant knowledge points and concepts
- Conservation of mechanical energy (no slipping, no dissipative work): initial potential energy mgh converts into translational and rotational kinetic energy.
- Moment of inertia of a solid sphere about its center: I = (2/5) mR^2.
- Rolling without slipping relation: v = ωR.

2) Complete reasoning process step by step
Start with energy conservation from rest at height h:
m g h = (1/2) m v^2 + (1/2) I ω^2.

Substitute I and ω = v/R:
I = (2/5) m R^2, ω = v/R,
so (1/2) I ω^2 = (1/2) * (2/5 m R^2) * (v^2 / R^2) = (1/5) m v^2.

Then total kinetic energy becomes:
(1/2) m v^2 + (1/5) m v^2 = (1/2 + 1/5) m v^2 = (5/10 + 2/10) m v^2 = (7/10) m v^2.

Thus m g h = (7/10) m v^2.

3) Intermediate algebra and solution for v and ω
Cancel m from both sides:
g h = (7/10) v^2  =>  v^2 = (10/7) g h  =>  v = sqrt((10/7) g h).

Angular speed:
ω = v / R = (1/R) sqrt((10/7) g h).

4) Why the correct answer is correct
From ω = (1/R) sqrt((10/7) g h) we see ω depends inversely on the radius R and on the drop height h, but the mass m has canceled out completely. Therefore ω depends on the radius of the sphere (and on h, which was not listed among choices) and does not depend on the mass.

5) Why the other options are incorrect
- "the mass of the sphere": incorrect because m cancels in the energy equation; ω is independent of m.
- "both the mass and the radius": incorrect because mass does not appear in the final expression.
- "neither the mass nor the radius": incorrect because ω contains an explicit 1/R factor.

6) Conclusion
The angular velocity at the bottom depends on the radius of the sphere. This matches the provided correct answer: the radius of the sphere.

Suppose a solid sphere of mass m and radius R rolls without slipping down an inclined plane starting from rest. The angular velocity of the sphere at the bottom of the incline depends on单项选择题

类似问题

A solid sphere of mass 1.0 kg and radius 0.010 m starts from rest and rolls without slipping down a 1.0 m high inclined plane. What is the speed of the sphere when it reaches the bottom of the inclined plane?

Suppose a solid sphere of mass m and radius R rolls without slipping down an inclined plane starting from rest. The angular velocity of the sphere at the bottom of the incline depends on

What are the formulas to find the acceleration of an object rolling without slipping on an inclined plane?

What is the velocity at the contact point between the object and the surface on rolling motion?

What is a rolling motion?

A disk and a sphere have same mass and are released simultaneously at the top of an inclined plane. They roll down without slipping. Which will reach the bottom first?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单