Question

A solid wheel with mass M, radius R, and rotational inertia MR2/2, rolls without sliding on a horizontal surface. A horizontal force F is applied to the axle and the center of mass has an acceleration a. The magnitudes of the applied force F and the frictional force f of the surface, respectively, are:

1. F=Ma,f=0

2. F=Ma,f=Ma/2

3. F=2Ma,f=Ma

4. F=2Ma,f=Ma/2

5. F =3Ma/2, f =Ma/2

Answer 1

Mass of the wheel = M

Radius of the wheel = R

Rotational inertia = I = MR²/2

Linear acceleration of the center of mass = a

Applied force = F

Thus,

For the linear acceleration,

F = M*a = Mg sin 90 - f    (1)          where F_s is the frictional force

For the angular acceleration,

I = R f                            (2)    where is the angular acceleration of the wheel

Since there is no slip, velocity and acceleration at the point of contact with the surface is zero at any given time. So,

= a / R                       (3)

From equation 2 and 3,

f = I / R

f = I a / R²                    

f= M R² a / 2 * R2

f = Ma / 2                 (4)

From equation (1),

F = Ma - 0 + f

F = Ma + Ma / 2

F = 3 Ma/2                      (5)

From equation 4 and 5,

magnitude of the applied force F and the frictional force f of the surface are,

F = 3Ma/2 , f = Ma/2

Answer is option 5.