A solid cylinder of mass 20kg rolls without slipping down a 30° slope. Find the acceleration and the frictional force needed to prevent slipping. (g=9.8m/s^2)

a = 3.27 m/s²

F = 32.7 N

Explanation:

Draw a free body diagram.  There are three forces:

Weight force mg pulling straight down.

Normal force N pushing perpendicular to the slope.

Friction force F pushing parallel up the slope.

Sum of forces in the parallel direction:

∑F = ma

mg sin θ − F = ma

Sum of torques about the cylinder's axis:

∑τ = Iα

Fr = ½ mr²α

F = ½ mrα

Since the cylinder rolls without slipping, a = αr.  Substituting:

F = ½ ma

Two equations, two unknowns (a and F).  Substituting the second equation into the first:

mg sin θ − ½ ma = ma

Multiply both sides by 2/m:

2g sin θ − a = 2a

Solve for a:

2g sin θ = 3a

a = ⅔ g sin θ

a = ⅔ (9.8 m/s²) (sin 30°)

a = 3.27 m/s²

Solving for F:

F = ½ ma

F = ½ (20 kg) (3.27 m/s²)

F = 32.7 N

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