Scalar & vector quantities

A racing car of mass m travels around a horizontal oval track. The curved sections of the track are semicircles.

At the instant shown in Figure 10 the car is moving with constant speed v in a circle of radius R.

The car has kinetic energy E_k. The resultant force acting on the car is F.

The maximum centripetal force that can be produced between the car’s tyres and the track is 24 kN. The minimum value of R is 230 m.

m = 1600 kg

The car just starts to slip when it travels on the track in a circular path of radius 230 m

Deduce the speed of the car.

The driver wishes to drive this car around the curved section of the track, without slipping, at a speed greater than your answer to Question 07.2.

Suggest one way in which the driver can achieve this.

Figure 11 shows the car on the curved section of a different oval track. The curved section of this track is a slope. This means that cars can travel at greater speeds than on the track shown in Figure 10.

The car in Figure 11 stays at the same height on the curved section of the track. There is no tendency for the car to slip up or down the track and therefore there is no sideways friction on the tyres.

Draw and label, on Figure 11, arrows to show the direction of each force that acts on the car as it travels around the curved section of the track.

A car has a greater maximum speed on a sloped track than on a horizontal track of the same radius.

Explain why.

A ramp is inclined at an angle θ to the horizontal. A block of weight 240 N is pushed up the ramp by a 25.0 N force. This force acts parallel to the ramp. The block experiences a frictional force of 3.0 N. The block moves at a constant speed.

What is θ ?

A cable system is to be used to transfer supplies across a river. A model of the proposed system is built in order to test its performance.

The model consists of a cable attached to two vertical posts A and B, as shown in Figure 3. A pulley P of negligible mass is attached to the cable.

In this question the length of the cable does not change and the weight of the cable can be ignored.

An object O is attached to P. In one test, O and P are at rest in the position shown in Figure 3. The weight of O is 350 N.

Figure 4 is a force diagram drawn to scale. It represents the magnitudes and directions of the tensions T₁ and T₂ in the cable when O is at rest in the position shown in Figure 3. At this position, resistive forces are zero.

Complete the force diagram. Go on to determine, using your diagram, the magnitudes of T₁ and T₂.

Figure 4

In a second test, pulley P with O attached is released from A. P and O move along the cable to B.

The change in height of the centre of mass of O between A and B is 4.5 m. The distance travelled along the cable is 18 m.

The speed of O when it reaches B is 6.5 m s⁻¹.

Calculate the average resistive force on O and P as they move from A to B.

O contains a fragile item packed in suitable material.

Explain how the material can prevent damage to the fragile item when O stops suddenly at B.

A force of magnitude F acts on a box of mass m that moves along a frictionless slope. The slope is at an angle α to the horizontal and the force acts at an angle β to the slope.

What is the magnitude of the acceleration of the block along the slope?

The weight of a uniform bar is W. An object also of weight W is attached to one end. The bar is pivoted at the other end and held horizontal by a rope attached to its centre. The tension in the rope is 4 W.

What is angle θ ?

Which is a pair of vectors?

A heavy uniform trapdoor is hinged to the floor. It is held open by a rope as shown.

Which arrow shows the direction of the reaction force of the hinge on the trapdoor?

A trolley moves down a slope with constant acceleration. The mass of the trolley is doubled and the trolley moves down the same slope again. Air resistance and friction are negligible.

Which is correct?

When an aircraft turns in a horizontal circular path, it banks at an angle θ.

The aircraft has mass m and travels at constant speed v in a horizontal circular path of radius r. The lift force acts at the angle θ.

What is tan θ ?

Figure 7a shows the front view of a vertical coil mounted on a circular frame.

Figure 7b is a side view showing a section through the frame and coil. A constant direct current in the coil produces magnetic flux represented by the magnetic field lines on this diagram.

Figure 7a Figure 7b

Point Q is at the centre of the coil. A sensor placed at Q detects B_H, the horizontal component of the magnetic flux density. The effect of the Earth’s magnetic field at Q is negligible.

Discuss whether a search coil is a suitable sensor to detect B_H.

B_H is measured at Q with the coil vertical.

The coil is now rotated about Q through 25° as shown in Figure 8. The current in the coil does not change.

A new measurement of B_H is made with the coil fixed in this new position.

Determine the percentage change in B_H produced by this rotation of the coil. Show your working.

Figure 9 shows a protractor being used to measure the angle through which the coil is rotated.

Estimate the percentage uncertainty in this result. Justify your answer.

Figure 10 shows an arrangement of two vertical coils. Four experiments are done using this arrangement.

Coil 1 and coil 2 are identical and have a radius r. The coils are separated by a distance r and have a common axis PR. Q is at the centre of coil 1.

The four different experiments investigate how B_H varies with x, the displacement of the sensor from Q along PR.

In experiment 1, the current in coil 1 is 225 mA and the current in coil 2 is zero.

In experiment 2, the current in coil 1 is zero and the current in coil 2 is 225 mA.

Figure 11 shows the results of experiment 1 and experiment 2.

During experiment 1, B_H is measured with the sensor at Q. The sensor is then moved along PR until the value of B_H is halved. The distance from Q to the sensor is x_0.5

In experiment 3, the current in both coils is 225 mA so that the magnetic fields produced by coil 1 and coil 2 are combined.

The resultant B_H has a constant maximum value in the region between x = and

Deduce, in mT, the value of B_H in this region.

State two characteristics of the magnetic field lines in this region.

In experiment 4, the current in coil 2 is reversed so that the direction of the magnetic field produced by coil 2 is also reversed. The magnitudes of the currents in coil 1 and coil 2 are still 225 mA.

Sketch a graph to show how B_H varies between x = 0 and x = r. The x-axis has been provided for you.

Your graph should include numerical values on your B_H axis that correspond to x = 0 and x = r.

Figure 3 shows a garden gate with a pulley system designed to close the gate.

The pulley system raises weight A when the gate is opened. When the gate is released, A falls. The horizontal cable C passes over pulley R. The tension in cable C causes the gate to close.

Weight A is a solid cylinder with the following properties:

length = 0.23 m weight = 35 N

Table 2 gives the density of three available materials.

Deduce which one of the three materials is used for A.

Figure 4 shows the pulley arrangement when the gate is closed.

Pulleys P and M are frictionless so that the tension in the rope attached to A is equal to the weight of A. A weighs 35 N and the weight of moveable pulley M is negligible.

Calculate the tension in the horizontal cable C when the gate is closed.

Pulley M is pulled to the left as the gate is opened.

Explain why this increases the tension in the horizontal cable C.

Figure 5 shows a plan view with the gate open. The horizontal cable C passes over pulley R and is attached to the door at D. The angle between the door and the horizontal cable C is 12°.

The horizontal distance between the hinge and D is 0.95 m.

The tension in the horizontal cable C is now 41 N.

Calculate the moment of the tension about the hinge.

The same system is attached to an identical gate with stiffer hinges. Now the system does not supply a sufficiently large moment to close the gate.

Discuss two independent changes to the design to increase the moment about the hinges due to horizontal cable C.

An aeroplane flies horizontally at 150 m s⁻¹ along a bearing 60° east of north.

How far north from its starting position is the aeroplane after one hour?

Figure 8 shows the H-shaped posts used in a game of rugby.

Figure 9 shows the path of a ball that is kicked and just passes over the crossbar. The initial velocity of the ball is 20.0 m s⁻¹ at an angle of 40.0° to the ground.

You should consider air resistance to be negligible and treat the ball as a simple projectile.

The top of the crossbar is 3.00 m above the horizontal ground.

Show that the minimum speed of the ball in flight is about 15 m s⁻¹.

Explain your answer.

The ball just passes over the crossbar at a time t after it is kicked.

Show that t must satisfy the following equation:

There are two solutions to the equation

Discuss which of the two solutions is the time taken for the ball to pass over the crossbar from when it is kicked.

In your answer you should

• state the value for t given by each solution

• explain the physical significance of the other solution.

Another attempt is made to kick the ball over the crossbar. The initial velocity of the ball is the same as in the first attempt. This kick is made from a horizontal distance of 38 m from the posts.

Deduce whether the ball can pass over the crossbar.

Figure 10 shows the variations with time of the vertical velocity of a ball with and without air resistance.

Discuss the features of the motion of the ball shown by the two graphs.

In your answer you should refer to

• the gradients of the graphs

• the area between each line and the time axis.

The diagram shows a vector diagram of two forces acting on an object. The diagram is drawn to scale. The magnitude of the smaller force is 5.0 N.

What is the magnitude of the resultant force on the object?

A pair of cameras is used on a motorway to help determine the average speed of vehicles travelling between the two cameras.

Figure 5 shows the speed–time graph for a car moving between the two cameras.

The speed limit for the motorway between the two cameras is 22 m s⁻¹.

Determine whether the average speed of the car exceeded this speed limit.

Markings called chevrons are used on motorways.

The chevron separation is designed to give a driver time to respond to any change in speed of the car in front. The driver is advised to keep a minimum distance d behind the car in front, as shown in Figure 6.

Government research suggests that the typical time for a driver to respond is between 1.6 s and 2.0 s.

Suggest a value for d where the speed limit is 31 m s⁻¹.

The chevron separation is based on the response time, not on the time taken for a car to stop.

The brakes of a car are applied when its speed is 31 m s⁻¹ and the car comes to rest. The total mass of the car is 1200 kg.

The average braking force acting on the car is 6.8 kN.

Calculate the time taken for the braking force to stop the car and the distance travelled by the car in this time.

Suggest why the chevron separation on motorways does not take into account the distance travelled as a car comes to rest after the brakes are applied.

At bends on motorways the road is sloped so that a car is less likely to slide out of its lane when travelling at a high speed.

Figure 7 shows a car of mass 1200 kg travelling around a curve of radius 200 m. The motorway is sloped at an angle of 5.0°.

Figure 8 shows the weight W and reaction force N acting on the car. The advisory speed for the bend is chosen so that the friction force down the slope is zero.

Suggest an appropriate advisory speed for this section of the motorway.

Safety barriers are used on UK motorways to prevent vehicles crossing from one carriageway to the other carriageway. The barriers also absorb some of the kinetic energy of a vehicle and deflect vehicles along the barrier.

The standard test of a safety barrier uses a vehicle that contains dummies. The total mass of the vehicle and its contents is 1.5 × 10³ kg and its initial speed is 110 km h^–1

Show that the initial kinetic energy of the test vehicle is 700 kJ.

The test vehicle hits a steel safety barrier at an angle of 20°, as shown in Figure 7.

Calculate the component of the momentum of the test vehicle in a direction along the line of the safety barrier. Give an appropriate unit for your answer.

Immediately after the collision, the test vehicle moves along the safety barrier with no change in its momentum in this direction.

Show that the kinetic energy lost in the collision is about 80 kJ.

The steel safety barrier deforms during the collision. For the barrier to pass the test, the test vehicle should not move more than 1.5 m towards the other carriageway.

The barrier can apply an average force of 60 kN at right angles to the carriageway.

Deduce whether the safety barrier will pass the test.

A different safety barrier uses a solid concrete wall which does not deform. The same standard test is carried out on a concrete wall.

Discuss which type of barrier would cause less damage to the dummies in the test.

Figure 8 shows a side view of an act performed by two acrobats. Figure 9 shows the view from above.

The acrobats, each of mass 85 kg, are suspended from ropes attached to opposite edges of a circular platform that is at the top of a vertical pole. The platform has a diameter of 2.0 m A motor rotates the platform so that the acrobats move at a constant speed in a horizontal circle, on opposite sides of the pole.

When the period of rotation of the platform is 5.2 s, the centre of mass of each acrobat is 5.0 m below the platform and the ropes are at an angle of 28.5° to the vertical as shown in Figure 8.

Show that the linear speed of the acrobats is about 4.5 m s⁻¹

Determine the tension in each rope that supports the acrobats.

Discuss the consequences for the forces acting on the pole when one acrobat has a much greater mass than the other.

Three coplanar forces F₁, F₂ and F₃ act on a point object.

Which combination of forces can never produce a resultant force of zero?

Figure 7

Explain how Figure 7 shows that the friction between the ladder and the wall is negligible.

The forces acting on the ladder are in equilibrium.

Draw an arrow on Figure 7 to show the direction of the resultant force from the ground acting on the ladder. Label your arrow G.

The ladder is 8.0 m long and weighs 390 N.

Calculate the magnitude of the resultant force from the wall on the ladder.

Suggest the changes to the forces acting on the ladder that occur when someone climbs the ladder .

Figure 8

The model consists of two identical trolleys of mass M on a ramp which is at 35° to the horizontal. The trolleys are connected by a wire that passes around a pulley of negligible mass at the top of the ramp.

Two concrete blocks each of mass m are loaded onto trolley A at the top of the ramp. The trolley is released and accelerates to the bottom of the ramp where it is stopped by a flexible buffer. The blocks are unloaded from trolley A and two blocks are loaded onto trolley B that is now at the top of the ramp. The trolleys are released and the process is repeated.

Figure 9 shows the side view of trolley A when it is moving down the ramp.

The tension in the wire when the trolleys are moving is T.

Draw and label arrows on Figure 9 to represent the magnitudes and directions of any forces and components of forces that act on trolley A parallel to the ramp as it travels down the ramp.

Assume that no friction acts at the axle of the pulley or at the axles of the trolleys and that air resistance is negligible.

Show that the acceleration a of trolley B along the ramp is given by

Compare the momentum of loaded trolley A as it moves downwards with the momentum of loaded trolley B.

In practice, for safety reasons there is a friction brake in the pulley that provides a resistive force to reduce the acceleration to 25% of the maximum possible acceleration.

The distance travelled for each journey down the ramp is 9.0 m.

The following data apply to the arrangement.

Mass of a trolley M = 95 kg Mass of a concrete block m = 30 kg

Calculate the time taken for a loaded trolley to travel down the ramp.

It takes 12 s to remove the blocks from the lower trolley and reload the upper trolley.

Calculate the number of blocks that can be transferred to the lower level in 30 minutes.