3.03n Equilibrium in 2D: particle under forces

3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of \(35 ^ { \circ }\) to the vertical, as shown in the diagram. Model the sign as a particle. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}

1. By resolving forces horizontally, show that the tension is the same in each string.
2. Find the tension in each string.
3. If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.

6. \begin{figure}[h] \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
The two blocks, \(A\) and \(B\), are modelled as particles.
Given that

• \(\tan \alpha = \frac { 5 } { 12 }\) and \(\tan \beta = \frac { 7 } { 24 }\) and \(\tan \gamma = \frac { 3 } { 4 }\)
• the coefficient of friction, \(\mu\), is the same between each block and its plane
• one of the blocks is on the point of sliding up its plane
• the tension in the string is \(T\)
  1. determine, in terms of \(m\) and \(g\), an expression for \(T\),
  2. draw a diagram showing the forces on block \(A\), clearly labelling each of the forces acting on the block,
  3. determine the value of \(\mu\), giving a justification for your answer. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_2266_50_312_1978}

18 An object, \(O\), of mass \(m\) kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at \(A\).
The longer string, of length 1.2 metres, is fixed to the ceiling at \(B\).
This object hangs 0.6 metres directly below the ceiling as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-28_252_940_667_552} 18

1. Show that the tension in the shorter string is over \(30 \%\) more than the tension in the longer string.
   18
2. The tension in the longer string is known to be \(2 g\) newtons. Find the value of \(m\).
   A rough wooden ramp is 10 metres long and is inclined at an angle of \(25 ^ { \circ }\) above the horizontal. The bottom of the ramp is at the point \(O\). A crate of mass 20 kg is at rest at the point \(A\) on the ramp.
   The crate is pulled up the ramp using a rope attached to the crate.
   Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-30_252_842_804_598}

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

10 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
3. Show that \(0.5 < \tan \alpha \leqslant 1\).
4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.

8 An object of weight 16 N is supported in equilibrium by a force of \(P \mathrm {~N}\) at \(30 ^ { \circ }\) to the vertical and by another of 10 N at \(\theta ^ { \circ }\) to the vertical as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85c5c346-8eb5-47ea-b94e-80b1a0038ce1-4_549_483_397_831}

1. Draw a triangle to show that the forces acting on the object are in equilibrium.
2. Find the two possible values of \(\theta\) and the corresponding values of \(P\).

11 Three light inextensible strings \(A C , C D\) and \(D B\), each of length 10 cm , are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-5_300_670_475_699} The ends \(A\) and \(B\) are fixed to points 20 cm apart on the same horizontal level. Two heavy particles, each of mass 2 kg , are attached at \(C\) and \(D\). The system remains in a vertical plane.

1. Determine the tension in each string.
2. The string \(C D\) is replaced by one of length \(L \mathrm {~cm}\), made of the same material. If the tension in \(A C\) is 50 N , show that \(L = 20 - 4 \sqrt { 21 }\).

\includegraphics{figure_1} A uniform ladder \(AB\), of length \(3a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(AC = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(DC\) is in the same vertical plane as the ladder \(AB\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \alpha = 2\tan \theta\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\). [9]

\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]

\includegraphics{figure_5} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(10\) N, \(50\) N and \(40\) N. The directions of the forces are as shown in the diagram.

1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
2. Given instead that \(F = 10\sqrt{2}\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force. [3]

\includegraphics{figure_3} Four coplanar forces of magnitude \(P\) N, 10 N, 16 N and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\). [6]

\includegraphics{figure_4} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(2F\) N, \(3F\) N and \(30\) N. The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]

\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N, 10 N and \(P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]

\includegraphics{figure_1} Coplanar forces of magnitudes \(P\) N, \(Q\) N, 16 N and 22 N act at a point in the directions shown in the diagram. The forces are in equilibrium. Find the values of \(P\) and \(Q\). [5]

\includegraphics{figure_3} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60°\) to a horizontal ceiling. The other string is attached at an angle \(α°\) to a vertical wall (see diagram). The tension in the string attached to the ceiling is 4 N. Find the tension in the string which is attached to the wall and find the value of \(α\). [6]

\includegraphics{figure_2} The diagram shows a smooth ring \(R\), of mass \(m\) kg, threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(AR\) of the string makes an angle of 30° with the vertical, the part \(BR\) makes an angle of 40° with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\). [5]

\includegraphics{figure_3} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.

1. Find the values of \(F\) and \(\theta\). [4]
2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces. [2]

\includegraphics{figure_5} A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T\) N acting at an angle of \(60°\) to the downward vertical (see diagram). The ring is in equilibrium.

1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R\) N and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\). [4]
2. The coefficient of friction between the rod and the ring is 0.7. Find the value of \(T\) for which the ring is about to slip. [3]

\includegraphics{figure_2} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45°\) to the horizontal and the tension in this string is 120 N. The other string is inclined at \(θ°\) to the horizontal and the tension in this string is \(T\) N. Find the values of \(T\) and \(θ\). [6]

A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4] \includegraphics{figure_1}

\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).

1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]

1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]

\includegraphics{figure_3} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is attached to two strings \(PA\) and \(PB\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(APB = 90°\) (see diagram). Find the tension in each of the strings. [5]

\includegraphics{figure_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).

1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]