Forces and equilibrium：練習問題

A particle of mass \(5 \text{ kg}\) is placed on a smooth horizontal surface. It is acted upon by two horizontal forces: \(15 \text{ N}\) to the right and \(7 \text{ N}\) to the left. Find the magnitude and direction of the resultant force acting on the particle.

A particle of mass \(4\text{ kg}\) is suspended in equilibrium by two light inextensible strings. One string is at an angle of \(30^\circ\) to the vertical and the other is at an angle of \(60^\circ\) to the vertical. Calculate the tension in the string making an angle of \(30^\circ\) with the vertical. (Take \(g = 10\text{ m s}^{-2}\)).

Three coplanar forces of magnitudes \(F\), \(2F\), and \(3F\) act at a point in directions with bearings of \(000^\circ\), \(120^\circ\), and \(240^\circ\) respectively. Calculate the magnitude of the resultant force.

A particle \(P\) of mass \(5\text{ kg}\) is connected to a particle \(Q\) of mass \(m\text{ kg}\) by a light inextensible string passing over a smooth pulley fixed at the edge of a table. \(P\) rests on the rough horizontal surface of the table with a coefficient of friction \(\mu = 0.4\). \(Q\) hangs vertically. Find the maximum value of \(m\) for which the system remains in equilibrium.

A bead of weight \(W\) is threaded on a smooth circular wire of radius \(r\) fixed in a vertical plane. The bead is held in equilibrium at a point where the radius makes an angle \(\alpha\) with the downward vertical by a horizontal force \(P\). Express \(P\) in terms of \(W\) and \(\alpha\).

A particle is in equilibrium under the action of three coplanar forces. Two of the forces are \(8 \text{ N}\) acting horizontally to the right and \(6 \text{ N}\) acting vertically upwards. Find the magnitude of the third force.

A block of mass \(8 \text{ kg}\) is placed on a rough plane inclined at \(25^\circ\) to the horizontal. The coefficient of friction between the block and the plane is \(0.4\). A force \(P \text{ N}\) acts upwards parallel to the plane. Given that the block is in equilibrium and on the point of sliding down the plane, calculate the magnitude of \(P\). (Take the acceleration due to gravity \(g = 10 \text{ m s}^{-2}\)).

A particle P of mass \(5 \text{ kg}\) rests on a rough plane inclined at \(30^\circ\) to the horizontal. The coefficient of friction between P and the plane is \(0.4\). A light inextensible string is attached to P, passes over a smooth fixed pulley at the top of the incline, and is attached to another particle Q of mass \(M \text{ kg}\), which hangs freely. Assuming \(g = 10 \text{ m s}^{-2}\), determine the range of values for \(M\) for which the system remains in equilibrium.

A particle of mass \(4 \text{ kg}\) is placed on a smooth plane inclined at an angle of \(30^\circ\) to the horizontal. The particle is held in equilibrium by a horizontal force of magnitude \(P \text{ N}\) acting in the vertical plane containing the line of greatest slope. Take \(g=10 \text{ m s}^{-2}\).

(a) Calculate the magnitude of the horizontal force \(P\).

(b) Calculate the magnitude of the normal reaction force acting on the particle.

A particle of mass \(4 \text{ kg}\) is placed on a rough plane inclined at an angle of \(20^\circ\) to the horizontal. The coefficient of friction between the particle and the plane is \(0.3\).

A horizontal force of magnitude \(P \text{ N}\) acts on the particle, keeping it in equilibrium. The force acts in the vertical plane containing the line of greatest slope.

(a) Find the value of \(P\) for which the particle is on the point of sliding down the plane.

(b) Find the value of \(P\) for which the particle is on the point of sliding up the plane.