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**

- Diagram and coordinate system
- Isolate the system
- Draw all forces acting
- Take components
- Apply
**F**=m**a**and/or**τ**= I**α**origin dependent! and constraints - Solve
- 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 F_{x} = mg sin θ
- f = m a_{x}

&sum τ_{z} = R f
= I_{cm} α = (2mR^{2}/5)(a_{x}/R)

Step 6: Solve:

From the torque equation, we obtain f = 2
m a_{x}/5. Substituting into the force equation gives: a_{x} = (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 F_{x} = mg
sin θ - f = m a_{x}

&sum τ_{z} = m g R sin θ
= I_{cp} α = (7mR^{2}/5)(a_{x}/R)

Step 6: Solve:

From the torque equation, we obtain immediately a_{x} = (5 g sin &theta)/7.

Lessons:

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