Forces, equilibrium and resultants

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.

Jackie says: "A person's weight on Earth is directly proportional to their mass." Tom says: "A person's weight on Earth is different to their weight on the moon." Only one of the statements below is correct. Identify the correct statement. Tick (✓) one box. [1 mark] Jackie and Tom are both wrong. \(\square\) Jackie is right but Tom is wrong. \(\square\) Jackie is wrong but Tom is right. \(\square\) Jackie and Tom are both right. \(\square\)

3 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_296_735_1685_705} Three horizontal forces of magnitudes \(150 \mathrm {~N} , 100 \mathrm {~N}\) and \(P \mathrm {~N}\) have directions as shown in the diagram. The resultant of the three forces is shown by the broken line in the diagram. This resultant has magnitude 120 N and makes an angle \(75 ^ { \circ }\) with the 150 N force. Find the values of \(P\) and \(\theta\).

\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]

3 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-2_597_616_888_762} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes \(6 \mathrm {~N} , 5 \mathrm {~N} , F \mathrm {~N}\) and \(F \mathrm {~N}\) acting in the directions shown. Find the values of \(\alpha\) and \(F\).

3 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-04_442_636_264_758} Coplanar forces of magnitudes \(30 \mathrm {~N} , 15 \mathrm {~N} , 33 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram, where \(\tan \alpha = \frac { 4 } { 3 }\). The system is in equilibrium.

1. Show that \(\left( \frac { 14.4 } { 30 - P } \right) ^ { 2 } + \left( \frac { 28.8 } { P + 30 } \right) ^ { 2 } = 1\).
2. Verify that \(P = 6\) satisfies this equation and find the value of \(\theta\).

\includegraphics{figure_2} A particle \(P\) of mass \(1.6\) kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20°\) and \(40°\) respectively with the horizontal (see diagram). Find the tensions in the two strings. [6]

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

6 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.

1. Given that \(X = 100\), find the tension in the lower string.
2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]

A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]

Two forces \(\begin{bmatrix}3\\-2\end{bmatrix}\) N and \(\begin{bmatrix}-7\\-5\end{bmatrix}\) N act on a particle. Find the resultant force. Circle your answer. [1 mark] \(\begin{bmatrix}-21\\10\end{bmatrix}\) N \(\begin{bmatrix}-4\\-7\end{bmatrix}\) N \(\begin{bmatrix}4\\3\end{bmatrix}\) N \(\begin{bmatrix}10\\7\end{bmatrix}\) N

A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t\) s after its release the velocity of \(P\) is 7.5 m s\(^{-1}\) downwards. A vertically downwards force of magnitude 0.6t N acts on \(P\). A vertically upwards force of magnitude \(ke^{-t}\) N, where \(k\) is a constant, also acts on \(P\).

1. Show that \(\frac{dv}{dt} = 10 - 5ke^{-t} + 3t\). [2]
2. Find the greatest value of \(k\) for which \(P\) does not initially move upwards. [3]
3. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]

7 An object of mass 5 kg has a constant acceleration of \(\binom { - 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) for \(0 \leqslant t \leqslant 4\), where \(t\) is the time in seconds.

1. Calculate the force acting on the object. When \(t = 0\), the object has position vector \(\binom { - 2 } { 3 } \mathrm {~m}\) and velocity \(\binom { 4 } { 5 } \mathrm {~ms} ^ { - 1 }\).
2. Find the position vector of the object when \(t = 4\).

1. A particle \(P\) of mass 0.5 kg moves under the action of a single constant force ( \(2 \mathbf { i } + 3 \mathbf { j }\) )N.
   1. Find the acceleration of \(P\).
   At time \(t\) seconds, \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = 4 \mathbf { i }\)
2. Find the speed of \(P\) when \(t = 3\) Given that \(P\) is moving parallel to the vector \(2 \mathbf { i } + \mathbf { j }\) at time \(t = T\)
3. find the value of \(T\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]

A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )

1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
   Given that \(\lambda = 2\)
2. find the length of \(A B\).

3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).

A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of \(10 \text{ m s}^{-1}\) The first force acting on the particle is \((400\mathbf{i} + 180\mathbf{j})\) N. The second force acting on the particle is \((p\mathbf{i} - 180\mathbf{j})\) N. Find the value of \(p\). Circle your answer. [1 mark] \(-400\) \quad \(-390\) \quad \(390\) \quad \(400\)

5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

3 A particle hangs at the end of a string. A horizontal force of magnitude \(F \mathrm {~N}\) acting on the particle holds it in equilibrium so that the string makes an angle of \(20 ^ { \circ }\) with the vertical, as shown in the diagram. The tension in the string is 12 N . \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-04_357_374_1409_239}

1. Find the value of \(F\).
2. Find the mass of the particle.

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 \(2 : 1\).

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 \(F\).

\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]

3 \includegraphics[max width=\textwidth, alt={}, center]{d5acfe31-8614-4508-ac5b-865e15a1f539-2_661_565_1069_790} A small smooth ring \(R\) of weight 8.5 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A horizontal force of magnitude 15.5 N acts on \(R\) so that the ring is in equilibrium with angle \(A R B = 90 ^ { \circ }\). The part \(A R\) of the string makes an angle \(\theta\) with the horizontal and the part \(B R\) makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is \(T \mathrm {~N}\). Show that \(T \sin \theta = 12\) and \(T \cos \theta = 3.5\) and hence find \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-04_337_661_262_740} A smooth ring \(R\) of mass 0.2 kg is threaded on a light string \(A R B\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(A B R = 90 ^ { \circ }\). The angle between the part \(A R\) of the string and the vertical is \(60 ^ { \circ }\). The ring is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), acting on the ring in a direction perpendicular to \(A R\) (see diagram). Calculate the tension in the string and the value of \(X\).

A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \text{ m s}^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is 2100 N, find the resistance to motion of the car. [3]

A trailer of mass \(600\) kg is attached to a car of mass \(1100\) kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8\) m s\(^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is \(700\) N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is \(2100\) N, find the resistance to motion of the car. [3]

\includegraphics{figure_4} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at 20 m s\(^{-1}\) as they enter a zone where the speed limit is 14 m s\(^{-1}\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s\(^{-1}\) is 100 m.

1. Find the deceleration of the truck and the car. [3]

The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos \(\theta = 0.9\), find

1. the force in the towbar, [4]
2. the value of \(R\). [4]

A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s\(^{-2}\), in a straight line, on a smooth horizontal surface. Find the weight of the crate. Circle your answer. [1 mark] 15 kg \quad 15g N \quad 15 N \quad 15g kg

A vehicle is driven at a constant speed of \(12\text{ ms}^{-1}\) along a straight horizontal road. Only one of the statements below is correct. Identify the correct statement. Tick (\(\checkmark\)) one box. The vehicle is accelerating The vehicle's driving force exceeds the total force resisting its motion The resultant force acting on the vehicle is zero The resultant force acting on the vehicle is dependent on its mass [1 mark]

A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v\,\text{m}\,\text{s}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).

1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{d}v}{\text{d}t} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]

\includegraphics{figure_2} The diagram shows three coplanar forces acting at the point \(O\). The magnitudes of the forces are \(6 \text{ N}\), \(8 \text{ N}\) and \(10 \text{ N}\). The angle between the \(6 \text{ N}\) force and the \(8 \text{ N}\) force is \(90°\). The forces are in equilibrium. Find the other angles between the forces. [4]

1 Three horizontal forces, acting at a single point, have magnitudes \(12 \mathrm {~N} , 14 \mathrm {~N}\) and 5 N and act along bearings \(000 ^ { \circ } , 090 ^ { \circ }\) and \(270 ^ { \circ }\) respectively. Find the magnitude and bearing of their resultant.

In this question use \(g = 10\) m s⁻². A man of mass 80 kg is travelling in a lift. The lift is rising vertically. \includegraphics{figure_14} The lift decelerates at a rate of 1.5 m s⁻² Find the magnitude of the force exerted on the man by the lift. [3 marks]

\includegraphics{figure_8} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15°\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find the tension in the rope. [2]

\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at 35° to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2. Find the set of values of \(m\) for which the two blocks remain at rest. [6]

Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]

A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\). The driving force produced by the engine is 1800 N.

1. Calculate the resistance to motion which the car experiences. [4 marks]

At the top of the hill, the road becomes horizontal.

1. Find the initial acceleration of the car. [3 marks]

1. In the case where \(F = 20\), find the tensions in each of the strings.
2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.

\includegraphics{figure_1} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground. [3]

A car of mass \(1800\text{ kg}\) is towing a trailer of mass \(400\text{ kg}\) along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5\text{ m s}^{-2}\). There are constant resistance forces of \(250\text{ N}\) on the car and \(100\text{ N}\) on the trailer.

1. Find the tension in the tow-bar. [2]
2. Find the power of the engine of the car at the instant when the speed is \(20\text{ m s}^{-1}\). [3]

The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}

1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
2. Find the acceleration of the ring when \(T = 3\). [3]

1 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h] \end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.

5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.

1. Find the acceleration of the system and the tension in the cable.
2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.