Torque and Rotational Dynamics

A figure skater is spinning at a constant rate. Which of the following statements must be true regarding their state of equilibrium?

A block is sliding down a frictionless ramp. The force of gravity acts at the center of the block. Which of the following statements is true?

Under what condition can rotational equilibrium exist independently of translational equilibrium?

According to Newton's first law for rotation, what condition must be met for a system to maintain a constant angular velocity?

An object has three torques acting on it: 10 Nm clockwise, 5 Nm counterclockwise, and 5 Nm clockwise. What is the net torque on the object, and is it in rotational equilibrium?

A seesaw has two people sitting on it. One person applies a torque of 20 Nm counterclockwise, and the other applies a torque of 15 Nm clockwise. To achieve rotational equilibrium, what additional torque is needed, and in what direction should it be applied?

A spinning wheel has a moment of inertia of $2 kg cdot m^2$. If a net torque of $10 N cdot m$ is applied to the wheel, what is its angular acceleration?

A motor applies a torque to a rotating shaft with a moment of inertia of $5 kg cdot m^2$. Over 2 seconds, the angular velocity of the shaft increases from $2 rad/s$ to $6 rad/s$. Assuming the torque is constant, what is the magnitude of the torque applied by the motor?

A force $\vec{F} = (2\hat{i} - 3\hat{j}) N$ is applied to a point with position vector $\vec{r} = (1\hat{i} + 2\hat{j}) m$ relative to the axis of rotation. What is the torque produced by this force?

Two forces are applied to a rod that is free to rotate about one end. Force $\vec{F_1} = (5\hat{i}) N$ is applied at a distance of $2 m$ from the pivot point, and force $\vec{F_2} = (-3\hat{j}) N$ is applied at a distance of $4 m$ from the pivot point. The position vectors are $\vec{r_1} = (2\hat{j}) m$ and $\vec{r_2} = (4\hat{i}) m$ respectively. What is the net torque on the rod?