力矩：练习题

A uniform plank \( AB \) has length \( 6 \text{ m} \) and mass \( 30 \text{ kg} \). It is supported in a horizontal position by a single pivot at its center. A child of mass \( 45 \text{ kg} \) sits at point \( A \). A second child of mass \( 20 \text{ kg} \) sits at a distance \( x \) from the center to balance the plank. Find the value of \( x \).

A non-uniform rod \( PQ \) has length \( 4 \text{ m} \) and mass \( 12 \text{ kg} \). The center of mass of the rod is at the point \( G \), where \( PG = 1.2 \text{ m} \). The rod is supported horizontally in equilibrium by two vertical props, one at \( P \) and one at \( Q \). Find the magnitude of the reaction force at the prop at \( Q \). Take \( g = 9.8 \text{ m s}^{-2} \).

A uniform ladder of mass \( M \) and length \( L \) rests against a smooth vertical wall with its lower end on rough horizontal ground. The ladder is inclined at an angle \( \alpha \) to the horizontal. If the coefficient of friction between the ladder and the ground is \( \mu \), what is the minimum value of \( \tan \alpha \) for the ladder to remain in equilibrium?

A uniform beam AB has mass 15 kg and length 4 m. The beam is hinged to a vertical wall at A and is held in a horizontal position by a light cable CD. Point C lies on the beam such that AC = 3 m, and point D is on the wall vertically above A. The cable CD makes an angle of 30° with the beam. A particle of mass M is attached to the beam at B. Given that the maximum tension the cable can sustain is 500 N, find the maximum value of M, to one decimal place, such that the beam remains in equilibrium. Take \( g = 9.8 \text{ m s}^{-2} \).

A uniform rod \( XY \) of length \( 5 \text{ m} \) and mass \( 15 \text{ kg} \) is held in a horizontal position by two vertical supports, one at the end \( X \) and one at a point \( Z \), where \( Z \) is \( 1 \text{ m} \) from the end \( Y \). A particle of mass \( M \text{ kg} \) is placed at point \( Y \).
(a) Explain why the reaction force at the support \( X \) is zero when the rod is on the point of tilting about \( Z \).
(b) Calculate the value of \( M \) required for the rod to be on the point of tilting.
Take \( g = 9.8 \text{ m s}^{-2} \).

A non-uniform rod \( AB \) of mass \( M \text{ kg} \) and length \( 6 \text{ m} \) is supported in a horizontal position by two vertical pillars at \( C \) and \( D \), where \( AC = 1 \text{ m} \) and \( DB = 1 \text{ m} \). The center of mass of the rod is at a distance \( \bar{x} \) from point \( A \).
When a mass of \( 12 \text{ kg} \) is hung from end \( A \), the rod is on the point of tilting about \( C \).
When a mass of \( 20 \text{ kg} \) is hung from end \( B \), the rod is on the point of tilting about \( D \).
(a) By formulating two equations for the moments in each scenario, show that \( M = 8 \).
(b) Determine the value of \( \bar{x} \).