6.02b Calculate work: constant force, resolved component

4

1. A large nail of mass 0.02 kg has been driven a short distance horizontally into a fixed block of wood, as shown in Fig. 4.1, and is to be driven horizontally further into the block. The wood produces a constant resistance of 2.43 N to the motion of the nail. The situation is modelled by assuming that linear momentum is conserved when the nail is struck, that all the impacts with the nail are direct and that the head of the nail never reaches the wood. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_279_711_482_676} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The nail is first struck by an object of mass 0.1 kg that is moving parallel to the nail with linear momentum of magnitude 0.108 Ns . The object becomes firmly attached to the nail.
   1. Calculate the speed of the nail and object immediately after the impact.
   2. Calculate the time for which the nail and object move, and the distance they travel in that time. On a second attempt to drive in the nail, it is struck by the same object of mass 0.1 kg moving parallel to the nail with the same linear momentum of magnitude 0.108 Ns . This time the object does not become attached to the nail and after the contact is still moving parallel to the nail. The coefficient of restitution in the impact is \(\frac { 1 } { 3 }\).
   3. Calculate the speed of the nail immediately after this impact.
2. A small ball slides on a smooth horizontal plane and bounces off a smooth straight vertical wall. The speed of the ball is \(u\) before the impact and, as shown in Fig. 4.2, the impact turns the path of the ball through \(90 ^ { \circ }\). The coefficient of restitution in the collision between the ball and the wall is \(e\). Before the collision, the path is inclined at \(\alpha\) to the wall. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_294_590_1804_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Write down, in terms of \(u , e\) and \(\alpha\), the components of the velocity of the ball parallel and perpendicular to the wall before and after the impact.
   2. Show that \(\tan \alpha = \frac { 1 } { \sqrt { e } }\).
   3. Hence show that \(\alpha \geqslant 45 ^ { \circ }\).

2 This question is about 'kart gravity racing' in which, after an initial push, unpowered home-made karts race down a sloping track. The moving karts have only the following resistive forces and these both act in the direction opposite to the motion.

• A force \(R\), called rolling friction, with magnitude \(0.01 M g \cos \theta \mathrm {~N}\) where \(M \mathrm {~kg}\) is the mass of the kart and driver and \(\theta\) is the angle of the track with the horizontal
• A force \(F\) of varying magnitude, due to air resistance

A kart with its driver has a mass of 80 kg .
One stretch of track slopes uniformly downwards at \(4 ^ { \circ }\) to the horizontal. The kart travels 12 m down this stretch of track. The total work done by the kart against both rolling friction and air resistance is 455 J .

1. Calculate the work done against air resistance.
2. During this motion, the kart's speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(v \mathrm {~ms} ^ { - 1 }\). Use an energy method to calculate \(v\). To reach the starting line, the kart (with the driver seated) is pushed up a slope against rolling friction and air resistance. At one point the slope is at \(5 ^ { \circ }\) to the horizontal, the air resistance is 15 N , the acceleration of the kart is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up the slope and the power of the pushing force is 405 W .
3. Calculate the speed of the kart at this point.

2 A car of mass 850 kg is travelling along a road that is straight but not level.
On one section of the road the car travels at constant speed and gains a vertical height of 60 m in 20 seconds. Non-gravitational resistances to its motion (e.g. air resistance) are negligible.

1. Show that the average power produced by the car is about 25 kW . On a horizontal section of the road, the car develops a constant power of exactly 25 kW and there is a constant resistance of 800 N to its motion.
2. Calculate the maximum possible steady speed of the car.
3. Find the driving force and acceleration of the car when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When travelling along the horizontal section of road, the car accelerates from \(15 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6.90 seconds with the same constant power and constant resistance.
4. By considering work and energy, find how far the car travels while it is accelerating. When the car is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a constant slope inclined at \(\arcsin ( 0.05 )\) to the horizontal, the driving force is removed. Subsequently, the resistance to the motion of the car remains constant at 800 N .
5. What is the speed of the car when it has travelled a further 105 m up the slope?

3

1. A car of mass 900 kg is travelling at a steady speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at arcsin 0.1 to the horizontal. The power required to do this is 20 kW . Calculate the resistance to the motion of the car.
2. A small box of mass 11 kg is placed on a uniform rough slope inclined at arc \(\cos \frac { 12 } { 13 }\) to the horizontal. The coefficient of friction between the box and the slope is \(\mu\).
   1. Show that if the box stays at rest then \(\mu \geqslant \frac { 5 } { 12 }\). For the remainder of this question, the box moves on a part of the slope where \(\mu = 0.2\).
      The box is projected up the slope from a point P with an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It travels a distance of 1.5 m along the slope before coming instantaneously to rest. During this motion, the work done against air resistance is 6 joules per metre.
   2. Calculate the value of \(v\). As the box slides back down the slope, it passes through its point of projection P and later reaches its initial speed at a point Q . During this motion, once again the work done against air resistance is 6 joules per metre.
   3. Calculate the distance PQ.

4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and water have total mass 20 kg . In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.

1. How much work is done to raise the pail from rest so that it is travelling upwards at \(0.5 \mathrm {~ms} ^ { - 1 }\) when at a distance of 4 m above its starting position?
2. What power is required to raise the pail at a steady speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) ? Jack falls over and hurts himself. He then slides down a hill.
   His mass is 35 kg and his speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(3 \mathrm {~ms} ^ { - 1 }\) while descending through a vertical height of 3 m .
3. How much work is done against friction? In Jack's further motion, he slides down a slope at an angle \(\alpha\) to the horizontal where \(\sin \alpha = 0.1\). The frictional force on him is now constant at 150 N . For this part of the motion, Jack's initial speed is \(3 \mathrm {~ms} ^ { - 1 }\).
4. How much further does he slide before coming to rest?

2

1. A small block of mass 25 kg is on a long, horizontal table. Each side of the block is connected to a small sphere by means of a light inextensible string passing over a smooth pulley. Fig. 2 shows this situation. Sphere A has mass 5 kg and sphere B has mass 20 kg . Each of the spheres hangs freely. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-3_487_1123_466_552} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Initially the block moves on a smooth part of the table. With the block at a point O , the system is released from rest with both strings taut.
   1. (A) Is mechanical energy conserved in the subsequent motion? Give a brief reason for your answer.
      (B) Why is no work done by the block against the reaction of the table on it? The block reaches a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at point P .
   2. Use an energy method to calculate the distance OP. The block continues moving beyond P , at which point the table becomes rough. After travelling two metres beyond P , the block passes through point Q . The block does 180 J of work against resistances to its motion from P to Q .
   3. Use an energy method to calculate the speed of the block at Q .
2. A tree trunk of mass 450 kg is being pulled up a slope inclined at \(20 ^ { \circ }\) to the horizontal. Calculate the power required to pull the trunk at a steady speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) against a frictional force of 2000 N .

4 A box of mass 16 kg is on a uniformly rough horizontal floor with an applied force of fixed direction but varying magnitude \(P\) N acting as shown in Fig. 4. You may assume that the box does not tip for any value of \(P\). The coefficient of friction between the box and the floor is \(\mu\). \begin{figure}[h] \end{figure} Initially the box is at rest and on the point of slipping with \(P = 58\).

1. Show that \(\mu\) is about 0.25 . In the rest of this question take \(\mu\) to be exactly 0.25 .
   The applied force on the box is suddenly increased so that \(P = 70\) and the box moves against friction with the floor and another horizontal retarding force, \(S\). The box reaches a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest after 5 seconds; during this time the box slides 3 m .
2. Calculate the work done by the applied force of 70 N and also the average power developed by this force over the 5 seconds.
3. By considering the values of time, distance and speed, show that an object starting from rest that travels 3 m while reaching a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 5 seconds cannot be moving with constant acceleration. The reason that the acceleration is not constant is that the retarding force \(S\) is not constant.
4. Calculate the total work done by the retarding force \(S\).

4 A uniform solid sphere, of mass 14 kg and radius 0.25 m , is rotating about a fixed axis which is a diameter of the sphere. A couple of constant moment 4.2 Nm about the axis, acting in the direction of rotation, is applied to the sphere.

1. Find the angular acceleration of the sphere. During a time interval of 30 seconds the sphere rotates through 7500 radians.
2. Find the angular speed of the sphere at the start of the time interval.
3. Find the angular speed of the sphere at the end of the time interval.
4. Find the work done by the couple during the time interval.

4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) under the action of a force \(F \mathrm {~N}\), where \(F = 2 - 8 v ^ { 2 }\). The displacement of the particle from O at time \(t\) seconds is \(x \mathrm {~m}\).

1. Formulate and solve a differential equation to show that \(v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)\).
2. Hence express \(F\) in terms of \(x\) and find, by integration, the work done in the first 2 m of the motion.
3. Formulate and solve a differential equation to show that \(v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)\).
4. Calculate \(v\) when \(t = 1\) and when \(t = 2\), giving your answers to four significant figures. Hence find the impulse of the force \(F\) over the interval \(1 \leqslant t \leqslant 2\).

4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion \(R \mathrm {~N}\). At time \(t \mathrm {~s}\) the velocity of the parachutist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance she has fallen is \(x \mathrm {~m}\). While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as \(R = k v ^ { 2 }\), where \(k\) is a constant. The terminal velocity of the parachutist in free-fall is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = \frac { g } { 40 }\).
2. Show that \(v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)\). When she has fallen 1800 m , she opens her parachute.
3. Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist. As the parachute opens, the resistance instantly changes and is now modelled as \(R = 90 v\).
4. Calculate her velocity just before opening the parachute, correct to four decimal places.
5. Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. A bead is threaded on a straight wire. The vector equation of the wire is

$$\mathbf { r } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf { F } = ( 7 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\) in moving the bead from \(A\) to \(B\).
(Total 4 marks)

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

A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force ( \(5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) ) N. The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).

1. In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upwards.

A small smooth ring of mass 0.1 kg is threaded onto a smooth horizontal wire which is parallel to \(( \mathbf { i } + 2 \mathbf { j } )\). The only forces acting on the ring are its weight, the normal reaction from the wire and a constant force \(( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) N. The ring starts from rest at the point \(A\) on the wire, whose position vector relative to a fixed origin is \(( 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\), and passes through the point \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
(6)

2. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle \(P\) of mass 2 kg . The particle is initially at rest at the point \(A\) with position vector \(( - 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }\). Four seconds later, \(P\) is at the point \(B\) with position vector \(( 6 \mathbf { i } - 5 \mathbf { j } + 8 \mathbf { k } ) \mathrm { m }\). Given that \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\), find

1. \(\mathbf { F } _ { 2 }\),
2. the work done on \(P\) as it moves from \(A\) to \(B\).

1. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle. \(\mathbf { F } _ { 1 }\) has magnitude 9 N and acts in the direction of \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } . \mathbf { F } _ { 2 }\) has magnitude 18 N and acts in the direction of \(\mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }\).

Find the total work done by the two forces in moving the particle from the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) to the point with position vector \(( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\).
(Total 6 marks)

1. A bead of mass 0.5 kg is threaded on a smooth straight wire. The only forces acting on the bead are a constant force ( \(4 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) ) N and the normal reaction of the wire. The bead starts from rest at the point \(A\) with position vector \(( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and moves to the point \(B\) with position vector \(( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).

Find the speed of the bead when it reaches \(B\).
(4)

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

A small bead of mass 0.5 kg is threaded on a smooth horizontal wire. The bead is initially at rest at the point with position vector \(( \mathbf { i } - 6 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(\mathbf { P } \mathrm { N }\) then acts on the bead causing it to move along the wire. The bead passes through the point with position vector ( \(7 \mathbf { i } - 14 \mathbf { j }\) ) m with speed \(2 \sqrt { 7 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { P }\) is parallel to ( \(6 \mathbf { i } + \mathbf { j }\) ), find \(\mathbf { P }\).
(6)

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

A bead \(P\) of mass 0.2 kg is threaded on a smooth straight horizontal wire. The bead is at rest at the point \(A\) with position vector \(( 4 \mathbf { i } - \mathbf { j } ) \mathrm { m }\). A force \(( 0.2 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it to the point \(B\) with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the speed of \(P\) at \(B\).

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

A bead \(P\) of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation \(\mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } ) + \lambda ( - 2 \mathbf { i } + 3 \mathbf { j } )\). The bead is initially at rest at the point \(A\) with position vector \(( - \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(( 0.5 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire to the point \(B\). At \(B\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal, \(x - y\) plane.]

A bead \(P\) of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation \(y = 2 x - 1\). The unit of length on both axes is the metre. Initially the bead is at rest at the point \(( a , b )\). A force \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire so that \(P\) passes through the point \(( 5,9 )\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(a\) and the value of \(b\).

1 A child is pulling a toy block in a straight line along a horizontal floor.
The block is moving with a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) by means of a constant force of magnitude 20 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal. The work done by the force in 10 s is 350 J . Calculate the value of \(\theta\).

5 In the diagram below, points \(\mathrm { A } , \mathrm { B }\) and C lie in the same vertical plane. The slope AB is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(\mathrm { AB } = 5 \mathrm {~m}\). The point B is a vertical distance of 6.5 m above horizontal ground. The point C lies on the horizontal ground. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-6_601_1285_395_244} Starting at A , a particle P , of mass \(m \mathrm {~kg}\), moves along the slope towards B , under the action of a constant force \(\mathbf { F }\). The force \(\mathbf { F }\) has a magnitude of 50 N and acts at an angle of \(\theta ^ { \circ }\) to AB in the same vertical plane as A and B . When P reaches \(\mathrm { B } , \mathbf { F }\) is removed, and P moves under gravity landing at C . It is given that

• the speed of P at A is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at B is \(6 \mathrm {~ms} ^ { - 1 }\),
• the speed of P at C is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• 58 J of work is done against non-gravitational resistances as P moves from A to B ,
• 42 J of work is done against non-gravitational resistances as P moves from B to C .
  1. By considering the motion from B to C, show that \(m = 4.33\) correct to 3 significant figures.
  2. By considering the motion from A to B , determine the value of \(\theta\).
  3. Calculate the power of \(\mathbf { F }\) at the instant that P reaches B .