Does a ball rolling on an incline have the same acceleration on the way up as it – [Free] B71
Does a ball rolling on an incline have the same acceleration on the way up as it does on the way down
Does a Ball Rolling on an Incline Have the Same Acceleration Uphill and Downhill?
Answer:
Yes, the ball has the same magnitude of acceleration when rolling uphill and downhill on an incline, assuming it rolls without slipping. The direction (or sign) of the acceleration simply reverses.
🔍 Physics Behind the Motion
Let’s consider a ball of mass m and radius R with a moment of inertia I = k m R², where k is a constant depending on the shape (e.g., k = 2/5 for a solid sphere).
⚙️ Rolling Without Slipping
For a ball rolling without slipping:
a = Rα
Where:
- a = translational acceleration
- α = angular acceleration
📐 Gravitational Component
Let the incline be at an angle θ. The component of gravitational force acting along the incline is:
F = m g sinθ
📘 Newton’s Second Law (Translation and Rotation)
For translation:
mg sinθ – f = ma
For rotation about the center of mass:
f R = I α = I (a / R)
Solving these together gives:
a = g sinθ / (1 + k)
⬇️ Downhill Motion
When rolling downhill, the acceleration is in the direction of motion:
a = +g sinθ / (1 + k)
⬆️ Uphill Motion
If rolling uphill (e.g., after a push), gravity opposes the motion, so we get:
a = −g sinθ / (1 + k)
Same magnitude, but the sign is reversed because the direction is opposite.
🌐 For a Solid Sphere
Using k = 2/5 for a solid sphere:
a = g sinθ / (1 + 2/5) = (5/7) g sinθ