The velocity of the sphere at the bottom of the incline is 2.85 m/s.
How to calculate the velocity of the sphere?
The velocity of the sphere at the bottom of the incline is calculated by applying the principle of conservation of energy as follows.
Kinetic energy at bottom = Potential energy at top
K.E(trans) + K.E(rotational) = mgh
¹/₂mv² + ¹/₂Iω² = mgLsinθ
where;
- I is the moment of inertia of the solid sphere
- L is the length of the incline
- ω is the angular speed
- m is the mass
- v is the linear speed
¹/₂mv² + ¹/₂(²/₅mr²)(v²/r²) = mgLsinθ
¹/₂mv² + ¹/₅(mv²) = mgLsinθ
¹/₂v² + ¹/₅v² = gLsinθ
⁷/₁₀ v² = gLsinθ
v² = (¹⁰/₇)gLsinθ
v = √ [ (¹⁰/₇)gLsinθ ]
The velocity of the sphere at the bottom of the incline is calculated as;
v = √ [ (¹⁰/₇)(9.8)(1.7)sin(20) ]
v = 2.85 m/s
The complete question is below:
An 7.80-cm-diameter, 320 g solid sphere is released from rest at the top of a 1.70-m-long, 20.0 ∘ incline. It rolls, without slipping, to the bottom. Find the speed of the sphere at the bottom of the incline.