Question

A wheel of radius r rolls to the right without slipping on a horizontal road. Its axle moves at a constant speed vaxle. 

(a) Find the velocities of points A, B, and C with respect to the axle. Express your answers in terms of vaxle and r, as needed. 

(b) Find the velocities of points A, B, and C with respect to the road. (c) Comment on the velocity of point C with respect to the road.

Answer 1

(a)

If the rotating frame is considered then the wheel will appear to rotate at its place with linear velocity vaxle. Thus, using angular velocity formula v = ω × r, where r is the radius of the wheel and ω is the angular velocity of the wheel which is constant.

 

Thus, v = ωr is the velocity for each and every point on the rim and the directions are indicated in the above diagram.

 

At point C, A, B, D the velocity will be vaxle = ωr from the rotating frame of reference.

 

(b)

If the velocity of each point on the rim of the wheel need to be calculated from the ground then each point have to be treated separately.

 

Point A is shown clearly in the below diagram.

 

 

It is easily seen that the linear velocity due to rotation and the axle velocity are in the same direction. It should be noted that from the last part velocity of the axle is same as the linear velocity of the wheel. This is the necessary condition for pure rotation.

vA = vaxle + v

vA = v + v

vA = 2V

 

Point B is shown clearly in the below diagram.

 

It should be noted that from the last part velocity of the axle is same as the linear velocity of the wheel

vB = √(vaxle)2 + (v)2

vB = √(v)2 + (v)2

vB = √2v

 

Point C is shown clearly in the below diagram.

 

It should be noted that from the last part velocity of the axle is same as the linear velocity of the wheel

vC = vaxle + v

vC = v – v

vC = 0

 

Point D is shown clearly in the below diagram.

 

It should be noted that from the last part velocity of the axle is same as the linear velocity of the wheel

vD = √(vaxle)2 + (v)2

vD = √(v)2 + (v)2

vD = √2v

 

The velocity at point C is zero for measuring from the ground and it is a non-zero value for measuring from the rotating frame itself. This is because in the ground frame the point of contact of the wheel with the ground has linear velocity of the wheel in the opposite direction to axle velocity which is in the rightward direction. Thus, both velocity cancels out and the resultant velocity at that point is zero. This ast first might seem non-logical but it should be noted that wheel moving or rotating right wards must have contact velocity zero or else the contact point will move forward with a nonzero velocity and the rest of the parts of the wheel will not and then pure rotation is not observed. Rather skidding is observed.

a. At point C, A, B, D the velocity will be vaxle = ωr from the rotating frame of reference.