運動、力與動量：練習題

Which of the following contains only vector quantities?

A sphere of mass \(0.50 \text{ kg}\) moving at \(4.0 \text{ m s}^{-1}\) collides with a stationary sphere of mass \(1.0 \text{ kg}\). After the collision, the \(0.50 \text{ kg}\) sphere continues in the same direction with a velocity of \(1.2 \text{ m s}^{-1}\).

Calculate the velocity of the \(1.0 \text{ kg}\) sphere immediately after the collision.

A non-uniform rod \(AB\) of length \(4.0 \text{ m}\) and weight \(100 \text{ N}\) is supported horizontally by two vertical supports at its ends \(A\) and \(B\). The reaction force at support \(A\) is \(60 \text{ N}\). How far is the centre of gravity of the rod from end \(A\)?

A vehicle starts from rest and travels along a straight road with a constant acceleration of \( 2.5 \text{ m s}^{-2} \) for a duration of \( 8.0 \text{ s} \). What is the total displacement of the vehicle during this time?

A car of mass \(1200\text{ kg}\) travels at a constant speed of \(15\text{ m s}^{-1}\) up a road inclined at \(5.0^\circ\) to the horizontal. The total resistive force acting on the car is \(400\text{ N}\). Calculate the useful power output of the car's engine.

Identify whether work done and momentum are scalar or vector quantities, and provide the reasoning for your classification of momentum.

A uniform rod is pivoted at its center. A downward force of \(12 \text{ N}\) is applied at a distance of \(0.40 \text{ m}\) to the left of the pivot. Calculate the magnitude of the force that must be applied downwards \(0.15 \text{ m}\) to the right of the pivot to maintain equilibrium.

A uniform wooden plank of length \(5.0 \text{ m}\) and mass \(30 \text{ kg}\) has a \(10 \text{ kg}\) block placed exactly at one of its ends. Calculate the distance of the new centre of gravity of the system from the end where the block is placed.

A uniform metre ruler of weight \(1.2 \text{ N}\) is pivoted at the \(50 \text{ cm}\) mark. A weight of \(2.0 \text{ N}\) is placed at the \(15 \text{ cm}\) mark. A second weight, \(W\), is placed at the \(85 \text{ cm}\) mark to keep the ruler in horizontal equilibrium.

a) State the principle of moments.
b) Calculate the magnitude of the weight \(W\) required to maintain equilibrium.

An electric crane lifts a bundle of steel beams with a total mass of \(1500 \text{ kg}\) through a vertical height of \(25 \text{ m}\) at a constant speed in \(12 \text{ s}\). The electrical power input to the crane is \(45 \text{ kW}\).

a) Calculate the work done by the crane in lifting the beams.
b) Determine the useful power output of the crane during the lift.
c) Calculate the efficiency of the crane system.