Scalars and vectors：練習問題

A uniform beam of length \( 4.0\text{ m} \) and weight \( 200\text{ N} \) is supported horizontally by two vertical wires attached at each end. A load of \( 600\text{ N} \) is placed on the beam at a distance of \( 1.0\text{ m} \) from the left end. What is the tension in the wire at the right end of the beam?

A car of mass \( 1200\text{ kg} \) accelerates from rest to a speed of \( 20\text{ m s}^{-1} \) in \( 8.0\text{ s} \). If the total resistive force acting on the car is a constant \( 400\text{ N} \), what is the average useful power output of the car's engine during this period?

An object is projected horizontally from the top of a cliff with a velocity of \( 15\text{ m s}^{-1} \). It hits the ground at a horizontal distance of \( 45\text{ m} \) from the base of the cliff. Assuming air resistance is negligible and using \( g = 9.81\text{ m s}^{-2} \), what is the vertical component of the velocity of the object just before it hits the ground?

A ball of mass \( m \) is dropped from a height \( h \) and rebounds to a height of \( 0.64h \). What is the magnitude of the impulse exerted by the floor on the ball? Take the acceleration due to gravity as \( g \).

Two trolleys, X and Y, are on a horizontal frictionless track. Trolley X has a mass of \( 2.0\text{ kg} \) and is moving at \( 4.0\text{ m s}^{-1} \) towards trolley Y, which has a mass of \( 1.0\text{ kg} \) and is stationary. The trolleys collide and stick together. What is the percentage of the initial total kinetic energy that is lost during the collision?

A uniform beam of mass \( 5.0 \text{ kg} \) and length \( 2.0 \text{ m} \) is supported at its center. Calculate the magnitude of the moment produced by a vertical force of \( 20 \text{ N} \) applied at one end of the beam.

A ball is kicked from the ground with an initial velocity of \( 20 \text{ m s}^{-1} \) at an angle of \( 30^{\circ} \) to the horizontal. Ignoring air resistance and using \( g = 9.81 \text{ m s}^{-2} \), determine the maximum height reached by the ball.

A block of mass \( 0.50 \text{ kg} \) slides down a rough slope inclined at \( 25^{\circ} \) to the horizontal. The block travels \( 3.0 \text{ m} \) along the slope at a constant velocity; calculate the work done against friction during this displacement.

A ball of mass 0.15 kg is dropped from a height of 2.0 m onto a hard floor. It rebounds to a height of 1.4 m.

(a) Calculate the speed of the ball just before it hits the floor and just after it leaves the floor. Assume air resistance is negligible. \( (g = 9.81 \text{ m s}^{-2}) \)
(b) Calculate the magnitude of the change in momentum of the ball during the impact.

A uniform wooden plank of length 4.0 m and mass 25 kg rests horizontally on two supports, A and B. Support A is at the left end of the plank, and support B is 1.0 m from the right end of the plank.

(a) State the principle of moments.
(b) Calculate the magnitude of the upward force exerted by support B on the plank when it is in equilibrium.
(c) A person of mass 70 kg starts walking from support A towards the right end. Determine the maximum distance the person can walk from support A before the plank starts to tip.