Particle with string at angle to wall

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\includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-06_438_463_264_840} A light string \(A B\) is fixed at \(A\) and has a particle of weight 80 N attached at \(B\). A horizontal force of magnitude \(P \mathrm {~N}\) is applied at \(B\) such that the string makes an angle \(\theta ^ { \circ }\) to the vertical (see diagram).

1. It is given that \(P = 32\) and the system is in equilibrium. Find the tension in the string and the value of \(\theta\).
2. It is given instead that the tension in the string is 120 N and that the particle attached at \(B\) still has weight 80 N . Find the value of \(P\) and the value of \(\theta\).

1. \begin{figure}[h] \end{figure} A particle \(P\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 12 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and \(O P\) making an angle of \(20 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the weight of \(P\).

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight \(W\) newtons is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 5 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and with \(O P\) making an angle of \(25 ^ { \circ }\) to the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the value of \(W\).

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude \(F\) newtons is applied to \(P\). The particle \(P\) is in equilibrium under gravity with the string making an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. Find, to 3 significant figures,

1. the tension in the string,
2. the value of \(F\).

1. \begin{figure}[h] \end{figure} A tennis ball \(P\) is attached to one end of a light inextensible string, the other end of the string being attached to a the top of a fixed vertical pole. A girl applies a horizontal force of magnitude 50 N to \(P\), and \(P\) is in equilibrium under gravity with the string making an angle of \(40 ^ { \circ }\) with the pole, as shown in Fig. 1. By modelling the ball as a particle find, to 3 significant figures,

1. the tension in the string,
2. the weight of \(P\).

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a force of magnitude \(F\) newtons. The direction of this force is perpendicular to the string and \(O P\) makes an angle of \(60 ^ { \circ }\) with the vertical, as shown in Figure 1. Find

1. the value of \(F\)
2. the tension in the string.

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light string. The other end of the string is attached to a fixed point \(O\). A force of magnitude \(F\) newtons is applied to \(P\). The line of action of the force is inclined to the horizontal at \(30 ^ { \circ }\) and lies in the same vertical plane as the string. The particle \(P\) is in equilibrium with the string making an angle of \(40 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the value of \(F\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 2.5 m . The other end of the string is attached to a fixed point \(A\) on a vertical wall. The tension in the string is 16 N . The particle is held in equilibrium by a force of magnitude \(F\) newtons, acting in the vertical plane which is perpendicular to the wall and contains the string. This force acts in a direction perpendicular to the string, as shown in Figure 2. Given that the horizontal distance of \(P\) from the wall is 1.5 m , find

1. the value of \(F\),
2. the value of \(m\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5f2d38d9-b719-4205-8cb0-caa959afc46f-16_186_830_292_557} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Two posts, \(A\) and \(B\), are fixed at the side of a straight horizontal road and are 816 m apart, as shown in Figure 3. A car and a van are at rest side by side on the road and level with \(A\). The car and the van start to move at the same time in the direction \(A B\). The car accelerates from rest with constant acceleration until it reaches a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The van accelerates from rest with constant acceleration for 12 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The van then moves at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car has been moving at \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 30 s , the van draws level with the car at \(B\), and each vehicle has then travelled a distance of 816 m .
   (a) Sketch, on the same diagram, a speed-time graph for the motion of each vehicle from \(A\) to \(B\).
   (b) Find the time for which the car is accelerating.
   (c) Find the value of \(V\).

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\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a light, inextensible string fixed at one end to a point \(P\). The other end is attached to a small object of weight 10 N . The object is subjected to a horizontal force \(H\) so that the string makes an angle of \(30 ^ { \circ }\) with the vertical.

1. Find the magnitude of the tension in the string.
2. Show that the ratio of the magnitude of the tension to the magnitude of \(H\) is \(2 : 1\).