A.3 功、能量與功率：練習題

A particle has kinetic energy \(E_k\) when its speed is \(v\). What is the kinetic energy of the particle when its speed is increased to \(3v\)?

An object of mass \(m\) is pulled up a rough incline of vertical height \(h\) and slope length \(L\) at a constant speed. If the constant frictional force acting on the object is \(f\), what is the work done by the pulling force?

A motor pulls a block of mass \(m\) up a frictionless ramp inclined at an angle \(\theta\) to the horizontal. The block starts from rest and is pulled with a constant acceleration \(a\). What is the instantaneous power delivered by the motor at time \(t\)?

The unit of power, the Watt (W), can be expressed in terms of SI base units. Which of the following is correct?

A spring with spring constant \(k\) is compressed by a distance \(x\), storing elastic potential energy \(E_p\). If the spring is then compressed by a total distance of \(2x\), what is the new elastic potential energy stored in the spring?

A crate is pushed with a horizontal force of \(50\text{ N}\) across a floor for a distance of \(10\text{ m}\). Calculate the work done by this force.

An electric motor with an efficiency of \(75\%\) lifts a mass of \(50\text{ kg}\) through a vertical height of \(12\text{ m}\) in \(10\text{ s}\). Calculate the electrical power input to the motor. (Use \(g = 9.8\text{ m s}^{-2}\)).

A small block slides down a frictionless track from a height \(H\) and enters a vertical circular loop of radius \(R\). Determine the minimum height \(H\) required so that the block does not lose contact with the track at the top of the loop.

An electric motor is used to raise a load of mass 50 kg vertically through a distance of 15 m in a time of 20 s. The load is moved at a constant speed.

(a) Calculate the gain in gravitational potential energy of the load.
(b) Determine the useful power output of the motor.
(c) The motor has an efficiency of 75%. Calculate the electrical power input to the motor.
(d) Calculate the total energy wasted as heat and sound during this process. (Use \( g = 9.81\text{ m s}^{-2} \))

An electric motor is used to lift a load of \(250\text{ kg}\) vertically at a constant speed of \(1.2\text{ m s}^{-1}\). The motor has an efficiency of \(65\%\). (Use \(g = 9.81\text{ m s}^{-2}\)).
(a) Calculate the useful power output of the motor.
(b) Determine the electrical power input required by the motor.
(c) If the motor operates for \(20\text{ s}\), calculate the total energy dissipated to the surroundings as thermal energy and sound.