Awheel rolls purely on ground. if the velocity of a point 'p as shown on periphery of a body has a velocity equal
to the velocity of the centre of mass of the body. then 0 = ?
(1) 120°
(3) 60
(2) 30°
(4) 90°

(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.  From Inscribed Angle Theorem, we know the inscribed angle is half the arc angle.  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°

d is the correct answer

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