Equilibrium on slope with horizontal force

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small block of weight 18 N is held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, by a force of magnitude \(P\) N. Find

1. the value of \(P\) when the force is parallel to the plane, as in Fig. 1,
2. the value of \(P\) when the force is horizontal, as in Fig. 2.

1 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small block of weight 12 N is at rest on a smooth plane inclined at \(40 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a force of magnitude \(P \mathrm {~N}\). Find the value of \(P\) when

1. the force is parallel to the plane as in Fig. 1,
2. the force is horizontal as in Fig. 2.

2 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_385_621_488_762} Small blocks \(A\) and \(B\) are held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Each is held in equilibrium by a force of magnitude 18 N . The force on \(A\) acts upwards parallel to a line of greatest slope of the plane, and the force on \(B\) acts horizontally in the vertical plane containing a line of greatest slope (see diagram). Find the weight of \(A\) and the weight of \(B\).

4 Fig. 4 shows a block of mass 15 kg on a smooth plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Calculate \(P\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a particle \(P\) of mass 4 kg on a smooth plane inclined at \(15 ^ { \circ }\) to the horizontal. \(P\) is held in equilibrium by a horizontal force, \(F\).

1. Show that the normal reaction exerted by the plane on \(P\) is 40.6 N correct to 3 significant figures.
2. Calculate the value of \(F\).

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass 5 kg lies on the surface of a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The particle is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The horizontal force acts in a vertical plane containing a line of greatest slope of the inclined plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the smallest possible value of \(H\). The horizontal force is now removed, and \(P\) starts to slide down the slope.
   In the first \(T\) seconds after \(P\) is released from rest, \(P\) slides 1.5 m down the slope.
2. Find the value of \(T\).

\includegraphics{figure_6} A particle of mass \(1.2\) kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The particle is kept in equilibrium by a horizontal force of magnitude \(P\) N acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is \(0.15\). Find the least possible value of \(P\). [6]

\includegraphics{figure_4} A uniform rod \(AB\) has weight \(15\) N and length \(1.2\) m. The end \(A\) of the rod is in contact with a rough plane inclined at \(30°\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).

1. Show that the magnitude of the force applied at \(B\) is \(4.33\) N, correct to \(3\) significant figures. [3]
2. Find the magnitude of the frictional force exerted by the plane on the rod. [2]
3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane. [3]

\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find

1. the value of \(P\), [4]
2. the value of \(\mu\). [5]

The horizontal force is removed.

1. Determine whether or not the parcel moves. [5]