平衡：練習題

A uniform plank of length \(4.0 \text{ m}\) and weight \(120 \text{ N}\) is supported by two vertical pillars, \(X\) and \(Y\). Pillar \(X\) is at one end of the plank and pillar \(Y\) is positioned \(1.0 \text{ m}\) from the opposite end. What is the upward force provided by pillar \(Y\)?

A uniform horizontal rod of weight \(20 \text{ N}\) is hinged to a vertical wall at point P and supported by a wire attached to the other end Q. The wire makes an angle of \(30^\circ\) with the rod. A load of \(50 \text{ N}\) is suspended from Q. What is the magnitude of the horizontal component of the force exerted by the hinge on the rod at P?

A uniform rod of length \(2.0 \text{ m}\) and weight \(50 \text{ N}\) is pivoted at one end. It is held in a horizontal position by a string attached to the other end. The string makes an angle of \(30^\circ\) with the rod. What is the tension in the string?

A picture of weight \(15 \text{ N}\) is suspended symmetrically from a nail by a cord of total length \(50 \text{ cm}\). The two ends of the cord are attached to the top corners of the picture, which are separated by a horizontal distance of \(40 \text{ cm}\). What is the tension in the cord?

A uniform horizontal beam of length \( 3.0 \, \text{m} \) and weight \( 120 \, \text{N} \) is pivoted at one end. It is supported by a vertical cable attached at a point \( 2.0 \, \text{m} \) from the pivot. Calculate the tension in the cable required to keep the beam in equilibrium.

(a) State the two conditions necessary for a rigid body to be in equilibrium.
(b) A uniform wooden plank of length \( 4.0\text{ m} \) and mass \( 22\text{ kg} \) is held in a horizontal position by two vertical ropes, \( A \) and \( B \). Rope \( A \) is attached to the left-hand end of the plank, and rope \( B \) is attached \( 0.50\text{ m} \) from the right-hand end. A toolbox of mass \( 12\text{ kg} \) is placed on the plank at a distance of \( 1.2\text{ m} \) from rope \( A \).
(i) By taking moments about the attachment point of rope \( A \), calculate the tension in rope \( B \).
(ii) Calculate the tension in rope \( A \).

A uniform rigid beam AB of length \( 4.0 \text{ m} \) and mass \( 15 \text{ kg} \) is hinged at point A to a vertical wall. The beam is held in a horizontal position by a steel cable attached at point B. The cable makes an angle of \( 35^\circ \) with the horizontal beam. A weight of \( 300 \text{ N} \) is suspended from the beam at a distance of \( 3.0 \text{ m} \) from the hinge A.

(a) By taking moments about point A, calculate the tension in the steel cable. Assume the acceleration of free fall \( g = 9.81 \text{ m s}^{-2} \).
(b) Calculate the magnitude of the horizontal component of the force exerted by the hinge on the beam at point A.
(c) Determine the magnitude and direction of the vertical component of the force exerted by the hinge at point A.