Sol. The figure shows the wheel rolling to the right from the dashed to the full position without slipping. The linear displacement of the center O is s, radial line CO rotates to the new position C ′ O ′ through the angle θ , where θ is measured from the vertical direction. If the wheel does not slip, the are CA must equal to distance s . Thus the displacement relationship and its two time derivatives give

s = r θ , v O = r ω ,a O = r α Ans

where v O = , a O = = , ω = , and α = = . The angle θ , of course, must be in radians. The acceleration a O will be directed in the sense opposite to that of v O if the wheel is slowing down. In this event, the angular acceleration α will have the sence opposite to that of ω .

The origin of fixed coordinates is taken arbitrarily but conveniently at the point of contact between C on the rim of the wheel and the ground. When point C has moved along its cycloidal path to C ′ , its new coordinates and their time derivatives become

x = s – r sin θ = r ( θ – sin θ )

y = r – r cos θ = r (1 – cos θ )

= (1 – cos θ ) = v 0 (1 – cos θ )

= r sin θ = v 0 sin θ

= (1 – cos θ ) + v O sin θ

= sin θ ) + v O cos θ

= a O (1 – cos θ ) r ω 2 sin θ

= a O sin θ + r ω 2 cos θ

For the desired instant of contact, θ = 0 and

= 0 and = r ω 2 Ans.

Thus, the acceleration of the point C on the rim at the instant of contact with the ground depends only on r and ω and is directed toward the center of the wheel. If desired, the velocity and acceleration of C at any position θ may be obtained by writing the expressions v = i + j and a = i + j.

Application of the kinematic relationships for a wheel which rolls without slipping should be recognized for various configurations of rolling wheels such as those illustrated on the right. If a wheel slips as it rolls, the foregoing relations are no longer valid.

Helpful Hints :

(i) These three relations are not entirely unfamiliar at this point, and their application to the rolling wheel should be mastered thoroughly.

(ii) Clearly, when θ = 0, the point of contact has zero velocity so that = = 0. The acceleration of the contact point on the wheel will also be obtained by the principles of relative motion.