A constant \(5.0\,\mathrm{N\,m}\) torque acts through \(4.0\,\mathrm{rad}\). Find the work done.

A motor delivers torque \(12\,\mathrm{N\,m}\) at angular speed \(8.0\,\mathrm{rad\,s^{-1}}\). Find the power.

A resisting torque acts opposite the angular displacement. Is its work positive or negative?

A wheel has \(I=0.60\,\mathrm{kg\,m^2}\) and \(\omega=10\,\mathrm{rad\,s^{-1}}\). Find \(K_{\mathrm{rot}}\).

A net torque of \(3.0\,\mathrm{N\,m}\) acts through \(12\,\mathrm{rad}\) on a wheel initially at rest with \(I=2.0\,\mathrm{kg\,m^2}\). Find its final angular speed.

A flywheel's rotational kinetic energy increases from \(40\,\mathrm{J}\) to \(95\,\mathrm{J}\). Find the net rotational work done.

A torque \(\tau(\theta)=2\theta\,\mathrm{N\,m}\) acts from \(\theta=0\) to \(\theta=3.0\,\mathrm{rad}\). Find the work done.

A wheel spins at \(20\,\mathrm{rad\,s^{-1}}\). A brake supplies constant opposing torque \(4.0\,\mathrm{N\,m}\) for \(5.0\,\mathrm{s}\). Find the average braking power during that interval if \(\omega\) decreases linearly to \(10\,\mathrm{rad\,s^{-1}}\).

A rotating disk with \(I=0.80\,\mathrm{kg\,m^2}\) speeds up from \(5.0\) to \(15\,\mathrm{rad\,s^{-1}}\). If a constant torque acts through \(20\,\mathrm{rad}\), find the torque.

A motor provides \(200\,\mathrm{W}\) to a wheel spinning at \(25\,\mathrm{rad\,s^{-1}}\). The load torque is \(5.0\,\mathrm{N\,m}\). Find the net torque available to accelerate the wheel.

A motor applies constant torque \(\tau_0\) to a drum of radius \(R\) and moment of inertia \(I\), lifting a mass \(m\) by a light cable without slip. Derive the upward acceleration, the condition for upward motion from rest, and the motor power as a function of angular speed.

A wheel starts from rest. A driving torque \(\tau(\theta)=\tau_0(1-\theta/\theta_0)\) acts for \(0\leq\theta\leq\theta_0\), while a constant resisting torque \(\tau_f\) opposes the motion. Derive the work accumulated by angle \(\theta\), the condition for the wheel to reach \(\theta_0\), and its angular speed there if it does.