Statics and forces: แบบฝึกหัด

A particle is in equilibrium under the action of three coplanar forces: \(\mathbf{F}_1 = (k\mathbf{i} + 5\mathbf{j}) \text{ N}\), \(\mathbf{F}_2 = (4\mathbf{i} - m\mathbf{j}) \text{ N}\), and \(\mathbf{F}_3 = (-7\mathbf{i} + 2\mathbf{j}) \text{ N}\). Determine the values of the constants \(k\) and \(m\).

A uniform rod of length \(2L\) and weight \(W\) rests in limiting equilibrium with one end on a rough horizontal floor and the other against a smooth vertical wall. The coefficient of friction between the floor and the rod is \(\mu\). If the rod makes an angle \(\theta\) with the horizontal floor, find the expression for \(\mu\) in terms of \(\theta\).

A particle of mass \(4 \text{ kg}\) rests on a smooth horizontal plane. Given that the acceleration due to gravity is \(g = 9.8 \text{ m s}^{-2}\), determine the magnitude of the normal reaction force exerted by the plane on the particle.

Two particles, \(A\) (mass \(3 \text{ kg}\)) and \(B\) (mass \(2 \text{ kg}\)), are connected by a light inextensible string which passes over a smooth fixed pulley. Assuming the system is released from rest, find the acceleration of particle \(A\). Use \(g = 9.8 \text{ m s}^{-2}\).

A uniform rod \(AB\) of length \(4 \text{ m}\) and weight \(W = 100 \text{ N}\) is held in horizontal equilibrium by a vertical force \(P\) acting upwards at end \(B\) and another vertical force \(Q\) acting upwards at a point \(C\), where the distance \(AC = 1 \text{ m}\). Determine the magnitude of the force \(P\).

A uniform horizontal beam \(AB\) has length \(6 \text{ m}\) and weight \(150 \text{ N}\). The beam is supported by two vertical pillars, one at point \(C\) where \(AC = 1 \text{ m}\) and the other at point \(D\) where \(DB = 1.5 \text{ m}\).

(a) Draw a force diagram showing all forces acting on the beam.

(b) Calculate the magnitudes of the forces exerted by the supports (the normal reactions) at \(C\) and \(D\).

A particle \(P\) of mass \(5 \text{ kg}\) is placed on a rough plane inclined at an angle of \(30^\circ\) to the horizontal. The coefficient of dynamic friction between the particle and the plane is \(0.4\). A light inextensible string attached to \(P\) pulls it parallel to the line of greatest slope up the plane with a constant tension \(T\). The particle accelerates up the plane at \(1.5 \text{ m s}^{-2}\).

Use \(g = 9.8 \text{ m s}^{-2}\).

(a) Determine the magnitude of the normal reaction force exerted by the plane on the particle.

(b) Calculate the magnitude of the tension \(T\) in the string.