3.03b Newton's first law: equilibrium

6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$

1. Find the velocity of the particle at the instant when its acceleration is zero.
   The particle comes to instantaneous rest at \(Q\).
2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
   1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
   3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) does not reach the pulley.]
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881} \(S _ { 1 }\) and \(S _ { 2 }\) are light inextensible strings, and \(A\) and \(B\) are particles each of mass 0.2 kg . Particle \(A\) is suspended from a fixed point \(O\) by the string \(S _ { 1 }\), and particle \(B\) is suspended from \(A\) by the string \(S _ { 2 }\). The particles hang in equilibrium as shown in the diagram.

1. Find the tensions in \(S _ { 1 }\) and \(S _ { 2 }\). The string \(S _ { 1 }\) is cut and the particles fall. The air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.2 N .
2. Find the acceleration of the particles and the tension in \(S _ { 2 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-2_520_719_1137_712} \(A\) and \(B\) are fixed points of a vertical wall with \(A\) vertically above \(B\). A particle \(P\) of mass 0.7 kg is attached to \(A\) by a light inextensible string of length \(3 \mathrm {~m} . P\) is also attached to \(B\) by a light inextensible string of length \(2.5 \mathrm {~m} . P\) is maintained in equilibrium at a distance of 2.4 m from the wall by a horizontal force of magnitude 10 N acting on \(P\) (see diagram). Both strings are taut, and the 10 N force acts in the plane \(A P B\) which is perpendicular to the wall. Find the tensions in the strings. [6]

1 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).

1 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_346_583_255_781} A small block of weight 5.1 N rests on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 8 } { 17 }\). The block is held in equilibrium by means of a light inextensible string. The string makes an angle \(\beta\) above the line of greatest slope on which the block rests, where \(\sin \beta = \frac { 7 } { 25 }\) (see diagram). Find the tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_307_857_1695_644} A particle \(P\) of mass 1.05 kg is attached to one end of each of two light inextensible strings, of lengths 2.6 m and 1.25 m . The other ends of the strings are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. \(P\) hangs in equilibrium at a point 1 m below the level of \(A\) and \(B\) (see diagram). Find the tensions in the strings.

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 \(B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .

1. Find the tension in the string.
2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
3. Find the elastic potential energy of the string in this position.
4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).

1 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_643_218_264_959} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length 0.8 m and modulus of elasticity 8 N . One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is 2 m vertically below \(A\). When the particle is in equilibrium the distance \(A P\) is 1.1 m (see diagram). Find the value of \(m\).

1 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_248_1267_276_440} A light elastic string has natural length 0.6 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.63 m apart. A particle \(P\) of mass 0.064 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.08 m below \(A B\) (see diagram). Find

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

5 One end of a light elastic string \(S _ { 1 }\) of modulus of elasticity 20 N and natural length 0.5 m is attached to a fixed point \(O\). The other end of \(S _ { 1 }\) is attached to a particle \(P\) of mass \(0.4 \mathrm {~kg} . P\) hangs in equilibrium vertically below \(O\).

1. Find the distance \(O P\). The opposite ends of a light inextensible string \(S _ { 2 }\) of length \(l \mathrm {~m}\) are now attached to \(O\) and \(P\) respectively. The elastic string \(S _ { 1 }\) remains attached to \(O\) and \(P\). The particle \(P\) hangs in equilibrium vertically below \(O\).
2. Find the tension in the inextensible string \(S _ { 2 }\) for each of the following cases:
   (a) \(l < 0.5\);
   (b) \(l > 0.6\);
   (c) \(l = 0.54\). In the case \(l = 0.54\), the inextensible string \(S _ { 2 }\) suddenly breaks and \(P\) begins to descend vertically.
3. Calculate the greatest speed of \(P\) in the subsequent motion.

3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m . The point \(A\) is 0.3 m above a smooth horizontal surface. The particle \(P\) moves in a horizontal circle on the surface with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Calculate the tension in the string. \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-05_67_1569_486_328}
2. Find the magnitude of the force exerted by the surface on \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).

1. Show that the tension in the string is 16 N .
2. Find the value of \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-3_451_432_1434_852} One end of a light inextensible string is attached to the highest point \(A\) of a solid fixed sphere with centre \(O\) and radius 0.6 m . The other end of the string is attached to a particle \(P\) of mass 0.2 kg which rests in contact with the smooth surface of the sphere. The angle \(A O P = 60 ^ { \circ }\) (see diagram). The sphere exerts a contact force of magnitude \(R \mathrm {~N}\) on \(P\) and the tension in the string is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(R + ( \sqrt { } 3 ) T = 4\). \(P\) is now set in motion, and moves with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on the surface of the sphere.
2. Find an equation involving \(R , T\) and \(\omega\).
3. Hence
   1. calculate \(R\) when \(\omega = 2\),
   2. find the greatest possible value of \(T\) and the corresponding speed of \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-2_892_412_1217_865} A hollow cylinder of radius 0.35 m has a smooth inner surface. The cylinder is fixed with its axis vertical. One end of a light inextensible string of length 1.25 m is attached to a fixed point \(O\) on the axis of the cylinder. A particle \(P\) of mass 0.24 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle, in contact with the inner surface of the cylinder, and with the string taut (see diagram).

1. Find the tension in the string.
2. Given that the magnitude of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the force exerted on \(P\) by the cylinder.

2 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_496_609_1535_769} One end of a light inextensible string of length 0.16 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(30 ^ { \circ }\) with the downward vertical through \(A\) (see diagram). \(P\) moves with constant speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the tension in the string,
2. the force exerted by the table on \(P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).

1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.

5 A small ball \(B\) of mass 0.2 kg is attached to fixed points \(P\) and \(Q\) by two light inextensible strings of equal length. \(P\) is vertically above \(Q\), the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical. \(B\) moves with constant speed in a horizontal circle of radius 0.6 m .

1. Given that the tension in the string \(P B\) is 7 N , calculate
   1. the tension in string \(Q B\),
   2. the speed of \(B\).
   3. Given instead that \(B\) is moving with angular speed \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the tension in the string \(Q B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, find the speed of \(P\).
2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,

1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
2. the magnitude of \(\mathbf { R }\),
3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).

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} Figure 1 shows a small smooth ring threaded onto a light inextensible string.
One end of the string is attached to a fixed point \(A\) on a horizontal ceiling and the other end of the string is attached to a fixed point \(B\) on the ceiling. A horizontal force of magnitude 2 N acts on the ring so that the ring rests in equilibrium at a point \(C\), vertically below \(B\), with the string taut. The line of action of the horizontal force and the string both lie in the same vertical plane. The angle that the string makes with the ceiling at \(A\) is \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The tension in the string is \(T\) newtons. The mass of the ring is \(M \mathrm {~kg}\).

1. Find the value of \(T\)
2. Find the value of \(M\)

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). Particle \(A\) is held at rest on a rough plane which is inclined to horizontal ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. Particle \(B\) hangs vertically below \(P\) with the string taut, at a height \(h\) above the ground, as shown in Figure 4. The part of the string between \(A\) and \(P\) lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
The coefficient of friction between \(A\) and the plane is \(\frac { 11 } { 36 }\) The particle \(A\) is released from rest and begins to move up the plane.

1. Show that the frictional force acting on \(A\) as it moves up the plane is \(\frac { 22 m g } { 39 }\)
2. Write down an equation of motion for \(B\).
3. Show that the acceleration of \(A\) immediately after its release is \(\frac { 1 } { 3 } g\) In the subsequent motion, \(A\) comes to rest before it reaches the pulley.
4. Find, in terms of \(h\), the total distance travelled by \(A\) from when it was released from rest to when it first comes to rest again.

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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 .
   1. Sketch, on the same diagram, a speed-time graph for the motion of each vehicle from \(A\) to \(B\).
   2. Find the time for which the car is accelerating.
   3. Find the value of \(V\).