Answer:
(1) 120°
Explanation:
The wheel is purely rolling, which means it rotates about an axis through the point where it touches the ground (see Figure 11-6).
The net velocity at point P on the periphery, relative to the bottom of the wheel, is:
v = ωr
where r is the distance from the point at the bottom of the wheel to P (see diagram).
To find r, we need to use some geometry. Â From Inscribed Angle Theorem, we know the inscribed angle is half the arc angle. Â And from Thales' Theorem, we know an angle inscribed across a diameter is a right angle.
Therefore:
cos (θ/2) = r / (2R)
r = 2R cos (θ/2)
So the net velocity at P is:
v = 2Rω cos (θ/2)
We want this to equal the velocity at the center of the circle, which is Rω.
Rω = 2Rω cos (θ/2)
1 = 2 cos (θ/2)
cos (θ/2) = 1/2
θ/2 = 60°
θ = 120°
