3.03d Newton's second law: 2D vectors

2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),

1. show that \(R = 860\),
2. find the tension in the tow-bar.

3. \begin{figure}[h] \end{figure} A car of mass 1200 kg moves along a straight horizontal road. In order to obey a speed restriction, the brakes of the car are applied for 3 s , reducing the car's speed from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brakes are then released and the car continues at a constant speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 s . Figure 2 shows a sketch of the speed-time graph of the car during the 7 s interval. The graph consists of two straight line segments.

1. Find the total distance moved by the car during this 7 s interval.
2. Explain briefly how the speed-time graph shows that, when the brakes are applied, the car experiences a constant retarding force.
3. Find the magnitude of this retarding force.

6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find

1. the magnitude of the acceleration of the van and the car,
2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.

8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find

1. the acceleration of the truck and the ear,
2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}

6. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 0.5 kg and \(m \mathrm {~kg}\) respectively, where \(m < 0.5\). The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially \(P\) is 3.15 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. After \(P\) has been descending for 1.5 s , it strikes the ground. Particle \(P\) reaches the ground before \(Q\) has reached the pulley.

1. Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string as \(P\) descends.
3. Show that \(m = \frac { 5 } { 18 }\).
4. State how you have used the information that the string is inextensible. When \(P\) strikes the ground, \(P\) does not rebound and the string becomes slack. Particle \(Q\) then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
5. Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.

3. A particle \(P\) of mass 0.4 kg moves under the action of a single constant force \(\mathbf { F }\) newtons. The acceleration of \(P\) is \(( 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the angle between the acceleration and \(\mathbf { i }\),
2. the magnitude of \(\mathbf { F }\). At time \(t\) seconds the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when \(t = 0 , \mathbf { v } = 9 \mathbf { i } - 10 \mathbf { j }\), (c) find the velocity of \(P\) when \(t = 5\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find

1. the acceleration of \(Q\),
2. the value of \(\mu\),
3. the tension in the string.
4. State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time \(t = 0\), a constant force \(\mathbf { F }\) of magnitude 4 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 3. The system moves under the action of this force until \(t = 6 \mathrm {~s}\). During the motion, the resistance to the motion of \(P\) has constant magnitude 1 N and the resistance to the motion of \(Q\) has constant magnitude 2 N . Find

1. the acceleration of the particles as the system moves under the action of \(\mathbf { F }\),
2. the speed of the particles at \(t = 6 \mathrm {~s}\),
3. the tension in the rod as the system moves under the action of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , \mathbf { F }\) is removed and the system decelerates to rest. The resistances to motion are unchanged. Find
4. the distance moved by \(P\) as the system decelerates,
5. the thrust in the rod as the system decelerates.

5. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\).

1. Show that the magnitude of the acceleration of \(P\) is \(2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the velocity of \(P\) at time \(t = 2\) seconds. Another particle \(Q\) moves with constant velocity \(\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
3. Find the distance moved by \(Q\) in 2 seconds.
4. Show that at time \(t = 3.5\) seconds both particles are moving in the same direction.

4. \begin{figure}[h] \end{figure} A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.

1. Find the acceleration of the lift.
2. Find the magnitude of the force exerted on the lift by the cable.

2. \begin{figure}[h] \end{figure} A vertical rope \(A B\) has its end \(B\) attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg , as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) using the rope \(A B\). The rope is modelled as a light inextensible string.

1. Find the tension in the rope \(A B\).
2. Find the magnitude of the force exerted on the scale pan by the brick.

7. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\). The force \(\mathbf { F } _ { 1 }\) is given by \(\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\).
Given that the resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ),

1. find \(\mathbf { F } _ { 2 }\) (7) The acceleration of \(P\) is \(( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the speed of \(P\) when \(t = 3\) seconds.
   

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively] Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\) of mass 0.5 kg . \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\).
Given that the resultant force of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is in the same direction as \(- 2 \mathbf { i } - \mathbf { j }\),

1. show that \(p - 2 q = - 16\) Given that \(q = 3\)
2. find the magnitude of the acceleration of \(P\),
3. find the direction of the acceleration of \(P\), giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

1. A truck of mass 1500 kg is moving on a straight horizontal road.

The engine of the truck is working at a constant rate of 30 kW .
The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the truck is moving at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the value of \(R\). Later on, the truck is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 8 }\) The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 500 N .
   The engine of the truck is again working at a constant rate of 30 kW . At the instant when the speed of the truck is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(V\)

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

A particle \(Q\) of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds ( \(t \geqslant 0\) ), the position vector of \(Q\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(Q\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) It is given that $$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$ where \(k\) is a constant.
Given that when \(t = 2\) particle \(Q\) is moving in the direction of the vector \(\mathbf { i } + \mathbf { j }\)

1. show that \(k = 4\)
2. find the magnitude of \(\mathbf { F }\) when \(t = 2\) Given that \(\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }\) when \(t = 0\)
3. find \(\mathbf { r }\) when \(t = 2\)

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

A particle \(P\) of mass 2 kg moves under the action of two forces, \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\).

1. Find the magnitude of the acceleration of \(P\). At time \(t = 0 , P\) has velocity ( \(- u \mathbf { i } + u \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) is a positive constant. At time \(t = T\) seconds, \(P\) has velocity \(( 10 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find
   1. the value of \(T\),
   2. the value of \(u\).

3. \begin{figure}[h] \end{figure} A lift of mass \(M \mathrm {~kg}\) is being raised by a vertical cable attached to the top of the lift. A person of mass \(m \mathrm {~kg}\) stands on the floor inside the lift, as shown in Figure 1. The lift ascends vertically with constant acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the cable is 2800 N and the person experiences a constant normal reaction of magnitude 560 N from the floor of the lift. The cable is modelled as being light and inextensible, the person is modelled as a particle and air resistance is negligible.

1. Write down an equation of motion for the person only.
2. Write down an equation of motion for the lift only.
3. Hence, or otherwise, find
   1. the value of \(m\),
   2. the value of \(M\).

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

A particle \(P\) of mass 2 kg moves under the action of two forces, \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) and \(( 2 q \mathbf { i } + p \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants. Given that the acceleration of \(P\) is \(( \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)

1. find the value of \(p\) and the value of \(q\).
2. Find the size of the angle between the direction of the acceleration and the vector \(\mathbf { j }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At \(t = T\) seconds, \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } - 13 \mathbf { j } )\).
3. Find the value of \(T\).

7. \begin{figure}[h] \end{figure} A simple lift operates by means of a vertical cable which is attached to the top of the lift. The lift has mass \(m\) A box \(Q\) is placed on the floor of the lift.
A box \(P\) is placed directly on top of box \(Q\), as shown in Figure 4.
The cable is modelled as being light and inextensible and air resistance is modelled as being negligible.
The tension in the cable is \(\frac { 42 m g } { 5 }\) The lift and its contents move vertically upwards with acceleration \(\frac { 2 g } { 5 }\) Using the model,

1. find, in terms of \(m\), the combined mass of boxes \(P\) and \(Q\) During the motion of the lift, the force exerted on box \(P\) by box \(Q\) is \(\frac { 14 m g } { 5 }\) Using the model,
2. find, in terms of \(m\), the mass of box \(P\)

3. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.] Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), are given by $$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$ where \(a\) and \(b\) are constants.
The forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\) of mass 4 kg .
Given that \(P\) rests in equilibrium on a smooth horizontal surface under the action of these three forces,

1. find the size of the angle between the direction of \(\mathbf { F } _ { 3 }\) and the direction of \(- \mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is now removed and replaced by the force \(\mathbf { F } _ { 4 }\) given by \(\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )\) N, where \(\lambda\) is a positive constant. When the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) act on \(P\), the acceleration of \(P\) has magnitude \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(\lambda\).
   

4. \begin{figure}[h] \end{figure} Figure 1 shows a large bucket used by a crane on a building site to move materials between the ground and the top of the building. The mass of the bucket is 15 kg . The bucket is attached to a vertical cable with the bottom of the bucket horizontal. The cable is modelled as light and inextensible. When the bucket is on the ground, a bag of cement of mass 25 kg is placed in the bucket. The bucket with the bag of cement moves vertically upwards with constant acceleration \(0.2 \mathrm {~ms} ^ { - 2 }\). Air resistance is modelled as being negligible.

1. Find the tension in the cable. At the top of the building, the bag of cement is removed. A box of tools of mass 12 kg is now placed in the bucket. Later on the bucket with the box of tools is moving vertically downwards with constant deceleration \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Air resistance is again modelled as being negligible.
2. Find the magnitude of the normal reaction between the bucket and the box of tools.

3. A tractor of mass 6 tonnes is dragging a large block of mass 2 tonnes along rough horizontal ground. The cable connecting the tractor to the block is horizontal and parallel to the direction of motion. The cable is modelled as being light and inextensible.
The driving force of the tractor is 7400 N and the resistance to the motion of the tractor is 200 N . The resistance to the motion of the block is \(R\) newtons, where \(R\) is a constant. Given that the tension in the cable is 6000 N and the tractor is accelerating,

1. find the value of \(R\).
2. State how you have used the fact that the cable is modelled as being inextensible.
   

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