13. A solid sphere is rolling without slipping along a horizontal surface with a speed of 5.50

m/s when it starts up a ramp that makes an angle of 25.0° with the horizontal. What is the

speed of the sphere after it has rolled 3.00 m up the ramp? Notice the sphere rises a

height h as it rolls up the ramp.

OF

VE

3.00 m

A) 4.01 m/s

B) 8.02 m/s

C) 1.91 m/s

D) 2.16 m/s

E) 3.53 m/s

E) 3.53 m/s

Explanation:

KE₁ + RE₁ = KE₂ + RE₂ + PE

½ mv₁² + ½ Iω₁² = ½ mv₂² + ½ Iω₂² + mgh

For a sphere, I = ⅖ mr²

½ mv₁² + ½ (⅖ mr²) ω₁² = ½ mv₂² + ½ (⅖ mr²) ω₂² + mgh

½ mv₁² + ⅕ mr² ω₁² = ½ mv₂² + ⅕ mr² ω₂² + mgh

Rolling without slipping means v = ωr.

½ mv₁² + ⅕ mv₁² = ½ mv₂² + ⅕ mv₂² + mgh

⁷/₁₀ mv₁² = ⁷/₁₀ mv₂² + mgh

⁷/₁₀ v₁² = ⁷/₁₀ v₂² + gh

Substitute:

⁷/₁₀ (5.50 m/s) ² = ⁷/₁₀ v₂² + (9.8 m/s²) (3.00 m sin 25.0°)

v₂ = 3.53 m/s