### Rolling Down an Incline

A solid sphere of radius R starts from rest at the top of an incline and rolls without slipping. What is its acceleration?

Use DID TASC
1. Diagram and coordinate system
2. Isolate the system
3. Draw all forces acting
4. Take components
5. Apply F=ma   and/or   τ = I α   origin dependent!  and constraints
6. Solve
7. Check!

Step 1: Draw diagram and define coordinate system(s).

Define the origin at the center of the sphere, with +x downslope and y perpendicular to the slope, and z into the screen.

Step 3: Draw all forces acting (see transparency).

Step 4: Take components (see transparency).

Step 5: Apply Newton's second law and constraints:

&sum Fx = mg sin θ - f = m ax
&sum τz = R f = Icm α = (2mR2/5)(ax/R)

Step 6: Solve:

From the torque equation, we obtain f = 2 m ax/5. Substituting into the force equation gives:   ax = (5 g sin &theta)/7.

OR......

Step 1: Draw diagram and define coordinate system(s).
Define the origin at the contact point, with +x downslope and y perpendicular to the slope, and z into the screen.

Step 3: Draw all forces acting (see transparency).

Step 4: Take components (see transparency).

Step 5: Apply Newton's second law and constraints:

&sum Fx = mg sin θ - f = m ax
&sum τz = m g R sin θ = Icp α = (7mR2/5)(ax/R)

Step 6: Solve:

From the torque equation, we obtain immediately ax = (5 g sin &theta)/7.

Lessons:

1. The acceleration for a rolling object is smaller than that of the same object that slides without friction down the incline.
2. In the torque equation, the source of torque and its magnitude can change if the origin is changed.
3. The acceleration must be independent of the coordinate system.