Question

A wheel with a mass of 10 kg and a diameter of 0.2m is released from rest on a 20 degree incline as shown. Determine the angular velocity of the wheel after travelling 0.5m down the incline.

Answer 1

Solution :

Given :

m = 10 kg

diameter of the wheel = 0.2 m

So, Radius of the wheel (r) = 0.1 m

.

And, Distance traveled along the incline : θ = 20^o

.

.

Here, h = d sinθ

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Now, Moment of inertia of the wheel is given as : I = m r²

^.

So, Total kinetic energy of the wheel will be : KE_total = KE_tran + KE_rot

∴ KE_total = (1/2) m v² + (1/2) I ω²

∴ KE_total = (1/2) m (ωr)² + (1/2)(m r²) ω²

∴ KE_total = m (ωr)²

.

Now, According to the conservation of energy : KE_total = ΔPE

∴ m ω² r² = m g h

∴ m ω² r² = m g d sinθ

∴ ω² r² = g d sinθ

∴ ω² (0.1 m)² = (9.81 m/s²)(0.5 m) sin(20)

∴ ω² = 167.76

∴ ω = 12.95 rad/s