Question

Two wheels with fixed hubs, each having a mass of 1kg , start from rest, and forces are applied: F1= 1 N to the first wheel and F2 to the second wheel. The first wheel has a radius of 0.5 m, and the second has a radius of 1 m. Both forces are applied tangentially to the rim of each wheel. Assume the hubs and spokes are massless, so that the moment of inertia about the center of mass is Icm= mR^2. How large must the force F2 be in order to impart identical angular cm accelerations on each wheel?

Answer 1

Answer:

2N

Explanation:

$m = m_1 = m_2 = 1 kg$

The torques generated by tangential forces on the 2 wheels are

$T_1 = F_1R_1 = 1*0.5 = 0.5 Nm$

$T_2 = F_2R_2 = F_2*1 = F_2 Nm$

According to Newton's 2nd law, the angular accelerations generated by these torque would be

$\alpha_1 = \frac{T_1}{I_1} = \frac{0.5}{mR_1^2} = \frac{0.5}{1*0.5^2} = 2 rad/s^2$

$\alpha_2 = \frac{T_2}{I_2} = \frac{F_2}{mR_2^2} = \frac{F_2}{1*1^2} = F_2 rad/s^2$

For the 2nd wheel to have the same angular acceleration, its force must be

$\alpha_1 = \alpha_2$

$2 = F_2$

$F_2 = 2N$