3.03d Newton's second law: 2D vectors

5 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal towbar. A resistance force of magnitude \(R\) newtons acts on the car and a resistance force of magnitude \(2 R\) newtons acts on the caravan. The car and caravan accelerate at a constant \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when a driving force of magnitude 4720 newtons acts on the car.

1. Find \(R\).
2. Find the tension in the towbar.

3 A box, of mass 3 kg , is placed on a rough slope inclined at an angle of \(40 ^ { \circ }\) to the horizontal. It is released from rest and slides down the slope.

1. Draw a diagram to show the forces acting on the box.
2. Find the magnitude of the normal reaction force acting on the box.
3. The coefficient of friction between the box and the slope is 0.2 . Find the magnitude of the friction force acting on the box.
4. Find the acceleration of the box.
5. State an assumption that you have made about the forces acting on the box.

4 A tractor, of mass 3500 kg , is used to tow a trailer, of mass 2400 kg , across a horizontal field. The trailer is connected to the tractor by a horizontal tow bar. As they move, a constant resistance force of 800 newtons acts on the trailer and a constant resistance force of \(R\) newtons acts on the tractor. A forward driving force of 2500 newtons acts on the tractor. The trailer and tractor accelerate at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. \(\quad\) Find \(R\).
2. Find the magnitude of the force that the tow bar exerts on the trailer.
3. State the magnitude of the force that the tow bar exerts on the tractor.

7 A particle, of mass 10 kg , moves on a smooth horizontal surface. A single horizontal force, \(( 9 \mathbf { i } + 12 \mathbf { j } )\) newtons, acts on the particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the acceleration of the particle.
2. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the velocity of the particle is \(( 2.2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the particle is at the origin.
   1. Find the distance between the particle and the origin when \(t = 5\).
   2. Express \(\mathbf { v }\) in terms of \(t\).
   3. Find \(t\) when the particle is travelling north-east.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-15_2484_1709_223_153}

3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.

1. \(\quad\) Find \(T\).
2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}

5 A block, of mass \(3 m\), is placed on a horizontal surface at a point \(A\). A light inextensible string is attached to the block and passes over a smooth peg. The string is horizontal between the block and the peg. A particle, of mass \(2 m\), is attached to the other end of the string. The block is released from rest with the string taut and the string between the peg and the particle vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-10_170_726_536_657} Assume that there is no air resistance acting on either the block or the particle, and that the size of the block is negligible. The horizontal surface is smooth between the points \(A\) and \(B\), but rough between the points \(B\) and \(C\). Between \(B\) and \(C\), the coefficient of friction between the block and the surface is 0.8 .

1. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(A\) and \(B\).
2. Given that the distance between the points \(A\) and \(B\) is 1.2 metres, find the speed of the block when it reaches \(B\).
3. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(B\) and \(C\).
4. Given that the distance between the points \(B\) and \(C\) is 0.9 metres, find the speed of the block when it reaches \(C\).
5. Explain why it is important to assume that the size of the block is negligible.
   [0pt] [1 mark]

2 Three forces \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } , ( p \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }\) and \(( - 8 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) act on a particle of mass 5 kg to produce an acceleration of \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). No other forces act on the particle.

1. Find the resultant force acting on the particle in terms of \(p\) and \(q\).
2. \(\quad\) Find \(p\) and \(q\).
3. Given that the particle is initially at rest, find the displacement of the particle from its initial position when these forces have been acting for 4 seconds.
   [0pt] [3 marks]

5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.

1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of the 7 kg particle just before it hits the ground.
3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
   [0pt] [4 marks]

6 A floor polisher consists of a heavy metal block with a polishing cloth attached to the underside. A light rod is also attached to the block and is used to push the block across the floor that is to be polished. The block has mass 5 kg . Assume that the floor is horizontal. Model the block as a particle. The coefficient of friction between the cloth and the floor is 0.2 .
A person pushes the rod to exert a force on the block. The force is at an angle of \(60 ^ { \circ }\) to the horizontal and the block accelerates at \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The diagram shows the block and the force exerted by the rod in this situation. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-14_309_205_772_1009} The rod exerts a force of magnitude \(T\) newtons on the block.

1. Find, in terms of \(T\), the magnitude of the normal reaction force acting on the block.
2. \(\quad\) Find \(T\).
   [0pt] [6 marks]

6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of \(1 \mathrm {~ms} ^ { - 2 }\). By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,

1. show that Corinne's mass is 44 kg ,
2. calculate the tension in the rope,
3. find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
4. Explain what effect this would have on the initial acceleration of the children.
   (1 mark)

8. \begin{figure}[h] \end{figure} Figure 3 shows two particles \(A\) and \(B\), of mass \(5 M\) and \(3 M\) respectively, attached to the ends of a light inextensible string of length 4 m . The string passes over a smooth pulley which is fixed to the edge of a rough horizontal table 2 m high. Particle \(A\) lies on the table at a distance of 3 m from the pulley, whilst particle \(B\) hangs freely over the edge of the table 1 m above the ground. The coefficient of friction between \(A\) and the table is \(\frac { 3 } { 20 }\). The system is released from rest with the string taut.

1. Show that the initial acceleration of the system is \(\frac { 9 } { 32 } \mathrm {~g} \mathrm {~ms} ^ { - 2 }\).
2. Find, in terms of \(g\), the speed of \(A\) immediately before \(B\) hits the ground. When \(B\) hits the ground, it comes to rest and the string becomes slack.
3. Calculate how far particle \(A\) is from the pulley when it comes to rest. END

1. \begin{figure}[h] \end{figure} Figure 1 shows a particle \(P\) of mass 4 kg on a smooth plane inclined at \(15 ^ { \circ }\) to the horizontal. \(P\) is held in equilibrium by a horizontal force, \(F\).

1. Show that the normal reaction exerted by the plane on \(P\) is 40.6 N correct to 3 significant figures.
2. Calculate the value of \(F\).

4. A lift of mass 70 kg is supported by a cable which remains taut at all times. A man of mass 90 kg gets into the lift and it begins to descend vertically from rest with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate, giving your answers correct to 3 significant figures,

1. the magnitude of the force which the lift exerts on the man,
2. the tension in the cable. Prior to slowing down, the lift is moving at \(2 \mathrm {~ms} ^ { - 1 }\). It then uniformly decelerates until it is brought to rest.
3. Find the impulse exerted by the cable on the lift in bringing the lift to rest.
4. Given that it takes 2 seconds to come to rest, use your answer to part (c) to calculate the magnitude of the force exerted by the cable on the lift in bringing the lift to rest.
   (2 marks)

2. A ball of mass 2 kg moves on a smooth horizontal surface under the action of a constant force, \(\mathbf { F }\). The initial velocity of the ball is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and 4 seconds later it has velocity \(( 10 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors.

1. Making reference to the mass of the ball and the force it experiences, explain why it is reasonable to assume that the acceleration is constant.
2. Find, giving your answers correct to 3 significant figures,
   1. the magnitude of the acceleration experienced by the ball,
   2. the angle which \(\mathbf { F }\) makes with the vector \(\mathbf { i }\).

7. A car of mass 1200 kg tows a trailer of mass 800 kg along a straight level road by means of a rigid towbar. The resistances to the motion of the car and the trailer are proportional to their masses. Given that the car experiences a resistance to motion of 300 N ,

1. find the resistance to motion which the trailer experiences. Given that the engine of the car exerts a driving force of 3 kN ,
2. find the acceleration of the system,
3. show that the tension in the towbar is 1200 N . When the system has reached a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it continues at constant speed until an electrical fault causes the engine of the car to switch off. The brakes are used to apply a constant retarding force until the system comes to rest. Given that the retarding force of the brakes has magnitude 1 kN and assuming that the original resistances to motion of the car and the trailer remain constant,
4. calculate the distance that the system travels during the braking period,
5. find the magnitude and direction of the force exerted by the towbar on the car.
6. Comment on the assumption that the original resistances to motion of the car and the trailer remain constant throughout the motion.

2. A particle of mass 8 kg moves in a horizontal plane and is acted upon by three forces \(\mathbf { F } _ { 1 } = ( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find the magnitude, in newtons, of the resultant force which acts on the particle, giving your answer in the form \(k \sqrt { } 5\).
2. Calculate, giving your answer in degrees correct to 1 decimal place, the angle the acceleration of the particle makes with the vector \(\mathbf { i }\).

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 14 }\). The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

1. show that the acceleration of the system is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
2. find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
3. Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
4. Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.

1. At time \(t = 0\), a particle of mass 2 kg has velocity \(( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors and \(\lambda > 0\).

Given that the speed of the particle at time \(t = 0\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. find the value of \(\lambda\). The particle experiences a constant retarding force \(\mathbf { F }\) so that when \(t = 5\), it has velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Show that \(\mathbf { F }\) can be written in the form \(\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) where \(\mu\) is a constant which you should find.
   (5 marks)
   

4. \begin{figure}[h] \end{figure} Figure 1 shows a weight \(A\) of mass 6 kg connected by a light, inextensible string which passes over a smooth, fixed pulley to a box \(B\) of mass 5 kg . There is an object \(C\) of mass 3 kg resting on the horizontal floor of box \(B\). The system is released from rest. Find, giving your answers in terms of \(g\),

1. the acceleration of the system,
2. the force on the pulley.
3. Show that the reaction between \(C\) and the floor of \(B\) is \(\frac { 18 } { 7 } \mathrm {~g}\) newtons.

3 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are
east, north and vertically upwards. \includegraphics[max width=\textwidth, alt={}, center]{cb72a1c4-f769-4348-ad7f-66c3c96e1732-3_401_686_368_721} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

• During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
• During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
• Alesha's mass, including her equipment, is 100 kg .
• Initially, her position vector is \(\left( \begin{array} { r } - 75 \\ 90 \\ 750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5 \\ 0 \\ - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
  1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of \(\left( \begin{array} { r } 0 \\ 0 \\ - 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0 \\ 0 \\ 880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50 \\ - 20 \\ 0 \end{array} \right) \mathrm { N }\) are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

1 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }\).

1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.

4 The three forces \(\left. \begin{array} { r } - 1 \\ 14 \\ - 8 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 3 \\ - 9 \\ 10 \end{array} \right) \mathrm { N }\) and \(\mathbf { F } \mathrm { N }\) act on a body of mass 4 kg in deep space and give it an acceleration of \(\left. \quad \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 2 }\).

1. Calculate \(\mathbf { F }\). At one instant the velocity of the body is \(\left. \begin{array} { r } - 3 \\ 3 \\ 6 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Calculate the velocity and also the speed of the body 3 seconds later.

5 The position vector of a toy boat of mass 1.5 kg is modelled as \(\mathbf { r } = ( 2 + t ) \mathbf { i } + \left( 3 t - t ^ { 2 } \right) \mathbf { j }\) where lengths are in metres, \(t\) is the time in seconds, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal, perpendicular unit vectors and the origin is O .

1. Find the velocity of the boat when \(t = 4\).
2. Find the acceleration of the boat and the horizontal force acting on the boat.
3. Find the cartesian equation of the path of the boat referred to \(x\) - and \(y\)-axes in the directions of \(\mathbf { i }\) and \(\mathbf { j }\), respectively, with origin O . You are not required to simplify your answer.

6 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 \(\binom { 4 } { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the position vector of the object when \(t = 4\).

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 \(t = 4\).