AP® Physics C: Mechanics · Unit 6 · Topic 6.5
Racing Objects Down a Ramp
A rolling object's speed at the bottom of a ramp depends only on how its mass is distributed, not on its total mass or radius. Objects that pack more mass near the axis (like solid spheres) win because less energy gets locked into spinning.
When something rolls without slipping down a ramp, gravitational potential energy splits into two pots: translational kinetic energy (1/2)mv2 from moving down the ramp, and rotational kinetic energy (1/2)Iw2 from spinning. The rolling constraint v = wR links the two speeds, so once you know the shape of the object, you know exactly how the energy divides between them.
Write the moment of inertia as I = kmR2, where k is a shape factor: 1/2 for a solid cylinder, 2/5 for a solid sphere, 1 for a thin hoop. Energy conservation gives mgh = (1/2)mv2 + (1/2)(kmR2)(v/R)2, and the mass and radius cancel completely. Solving gives v = √(2gh/(1+k)). A smaller k means less energy diverted into rotation, so more is left for forward motion, and a higher final speed. That is why a solid sphere (k=2/5) beats a solid cylinder (k=1/2), which beats a hoop (k=1). The same shape factor also sets the acceleration down an incline: a = g sin(theta)/(1+k).
This connects directly to rotational kinetic energy and to the rolling without slipping condition from earlier in the unit, since both depend on the same v = wR relationship. It also sets up angular momentum conservation problems, where objects with different k values respond differently to torques and collisions.
Key terms
- shape factor (k)
- The constant in I = kmR2 that depends only on an object's mass distribution, not its size or total mass.
- rolling without slipping
- Motion where the contact point has zero velocity, linking translational and angular speed by v = wR.
- rotational kinetic energy
- Energy of spinning motion, equal to (1/2)Iw2.
- moment of inertia (I)
- A measure of how an object's mass is distributed relative to its rotation axis, resisting changes in angular velocity.
Watch it explained
- How it worksHow does the split between rotational and translational energy decide which shape wins?
- Worked exampleFind the speed at the bottom of a 2 m high ramp for a solid cylinder that rolls from rest.
- Worked exampleFind the acceleration of a solid sphere rolling without slipping down a 20 degree incline.
- Common mistakeWhy doesn't the mass or the radius of a rolling object change its finishing speed?
Check yourself
End of topic.