Compute the angular velocity of the wheel:

A wheel of 0.5 m radius rolls over a horizontal surface without slipping as illustrated in Figure (a). The velocity of its centre is 2 m/sec and is constant. Compute the angular velocity of the wheel and velocities at C, D and E.

Solution

As the wheel rolls on the ground and does not slip or slide on the ground, the velocity of the point of contact of wheel with ground is same but the velocity of ground should be equal to zero.

∴          we get v_B  = 0

Now consider the wheel as the rigid body and the body rotating about an axis through the pole A and B is the point on the rim of the wheel.

Considering this motion as a plane motion, we have

v _B  = v _A  + v _AB  = 0

∴ v _A  = - v _AB

 = - A B . ω_i acting at right angles to BA in the opposite direction of v_A

∴  v _A /AB  = ω = 2 /0.5= 4 rad / sec. (numerically)

Now, refer Figure (b)

 v_C  = v _A  + v _AC

= 2.82 m / sec

Here,

θ = 45^o

Likewise, refer Figure (c)

v _D  = v _A  + v _AD

  = 4 m / sec

θ = 0^o

and v_E  = 2.82 m / sec

θ = 45^o