Rotating Systems: Energy & Momentum

A constant torque of 5 Nm is applied to a wheel. If the wheel rotates through an angle of $\pi$ radians, what is the work done by the torque?

The torque acting on a rotating object varies with angular position as $\tau(\theta) = 3\theta^2 + 2\theta$. What is the work done by the torque as the object rotates from $\theta = 0$ to $\theta = \pi$?

A rotating wheel connected to a spring is initially at rest. As the wheel rotates, it compresses the spring, storing potential energy. If the wheel's initial kinetic energy is 200 J and the spring stores 150 J of potential energy when the wheel stops, how much energy was lost due to friction?

A disc rotates from rest due to a constant applied torque. After some time, its rotational kinetic energy is $KE_1$. If the applied torque is doubled, what is the rotational kinetic energy of the disc after the same amount of time?

A constant torque of 12 Nm is applied to a door. The force is applied 0.5 m from the hinges. If the door rotates through 90 degrees, how much work is done?

A variable torque is given by $\tau(\theta) = A\sin(\theta)$, where A is a constant. Calculate the work done as the object rotates from $\theta = 0$ to $\theta = \frac{\pi}{2}$.

A torque vs. angular displacement graph shows a triangle with a base of 4 radians and a height of 8 Nm. What is the work done by the torque?

A rotating object has its rotational kinetic energy increased by 50 J due to an applied torque. According to the work-energy theorem, how much work was done by the torque?

A graph of torque vs. angular displacement is a rectangle with a height of 10 Nm and a width of 2 radians. What is the work done by the torque?

A solid cylinder is rolling without slipping and experiences a torque that does 100 J of work. If the cylinder's translational kinetic energy increases by 40 J, what is the increase in its rotational kinetic energy?