SL Paper 2
A vertical wall carries a uniform positive charge on its surface. This produces a uniform horizontal electric field perpendicular to the wall. A small, positively-charged ball is suspended in equilibrium from the vertical wall by a thread of negligible mass.
The charge per unit area on the surface of the wall is σ. It can be shown that the electric field strength E due to the charge on the wall is given by the equation
.
Demonstrate that the units of the quantities in this equation are consistent.
The thread makes an angle of 30° with the vertical wall. The ball has a mass of 0.025 kg.
Determine the horizontal force that acts on the ball.
The charge on the ball is 1.2 × 10−6 C. Determine σ.
The centre of the ball, still carrying a charge of , is now placed from a point charge Q. The charge on the ball acts as a point charge at the centre of the ball.
P is the point on the line joining the charges where the electric field strength is zero.
The distance PQ is .
Calculate the charge on Q. State your answer to an appropriate number of significant figures.
A sample of vegetable oil, initially in the liquid state, is placed in a freezer that transfers thermal energy from the sample at a constant rate. The graph shows how temperature of the sample varies with time .
The following data are available.
Mass of the sample
Specific latent heat of fusion of the oil
Rate of thermal energy transfer
Calculate the thermal energy transferred from the sample during the first minutes.
Estimate the specific heat capacity of the oil in its liquid phase. State an appropriate unit for your answer.
The sample begins to freeze during the thermal energy transfer. Explain, in terms of the molecular model of matter, why the temperature of the sample remains constant during freezing.
Calculate the mass of the oil that remains unfrozen after minutes.
Two players are playing table tennis. Player A hits the ball at a height of 0.24 m above the edge of the table, measured from the top of the table to the bottom of the ball. The initial speed of the ball is 12.0 m s−1 horizontally. Assume that air resistance is negligible.
The ball bounces and then reaches a peak height of 0.18 m above the table with a horizontal speed of 10.5 m s−1. The mass of the ball is 2.7 g.
Show that the time taken for the ball to reach the surface of the table is about 0.2 s.
Sketch, on the axes, a graph showing the variation with time of the vertical component of velocity vv of the ball until it reaches the table surface. Take g to be +10 m s−2.
The net is stretched across the middle of the table. The table has a length of 2.74 m and the net has a height of 15.0 cm.
Show that the ball will go over the net.
Determine the kinetic energy of the ball immediately after the bounce.
Player B intercepts the ball when it is at its peak height. Player B holds a paddle (racket) stationary and vertical. The ball is in contact with the paddle for 0.010 s. Assume the collision is elastic.
Calculate the average force exerted by the ball on the paddle. State your answer to an appropriate number of significant figures.
A pipe is open at both ends. A first-harmonic standing wave is set up in the pipe. The diagram shows the variation of displacement of air molecules in the pipe with distance along the pipe at time t = 0. The frequency of the first harmonic is f.
A transmitter of electromagnetic waves is next to a long straight vertical wall that acts as a plane mirror to the waves. An observer on a boat detects the waves both directly and as an image from the other side of the wall. The diagram shows one ray from the transmitter reflected at the wall and the position of the image.
An air molecule is situated at point X in the pipe at t = 0. Describe the motion of this air molecule during one complete cycle of the standing wave beginning from t = 0.
The speed of sound c for longitudinal waves in air is given by
where ρ is the density of the air and K is a constant.
A student measures f to be 120 Hz when the length of the pipe is 1.4 m. The density of the air in the pipe is 1.3 kg m–3. Determine, in kg m–1 s–2, the value of K for air.
Demonstrate, using a second ray, that the image appears to come from the position indicated.
Outline why the observer detects a series of increases and decreases in the intensity of the received signal as the boat moves along the line XY.
A student uses a load to pull a box up a ramp inclined at 30°. A string of constant length and negligible mass connects the box to the load that falls vertically. The string passes over a pulley that runs on a frictionless axle. Friction acts between the base of the box and the ramp. Air resistance is negligible.
The load has a mass of 3.5 kg and is initially 0.95 m above the floor. The mass of the box is 1.5 kg.
The load is released and accelerates downwards.
Outline two differences between the momentum of the box and the momentum of the load at the same instant.
The vertical acceleration of the load downwards is 2.4 m s−2.
Calculate the tension in the string.
Show that the speed of the load when it hits the floor is about 2.1 m s−1.
The radius of the pulley is 2.5 cm. Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.
After the load has hit the floor, the box travels a further 0.35 m along the ramp before coming to rest. Determine the average frictional force between the box and the surface of the ramp.
The student then makes the ramp horizontal and applies a constant horizontal force to the box. The force is just large enough to start the box moving. The force continues to be applied after the box begins to move.
Explain, with reference to the frictional force acting, why the box accelerates once it has started to move.
A girl rides a bicycle that is powered by an electric motor. A battery transfers energy to the electric motor. The emf of the battery is 16 V and it can deliver a charge of 43 kC when discharging completely from a full charge.
The maximum speed of the girl on a horizontal road is 7.0 m s–1 with energy from the battery alone. The maximum distance that the girl can travel under these conditions is 20 km.
The bicycle and the girl have a total mass of 66 kg. The girl rides up a slope that is at an angle of 3.0° to the horizontal.
The bicycle has a meter that displays the current and the terminal potential difference (pd) for the battery when the motor is running. The diagram shows the meter readings at one instant. The emf of the cell is 16 V.
The battery is made from an arrangement of 10 identical cells as shown.
Show that the time taken for the battery to discharge is about 3 × 103 s.
Deduce that the average power output of the battery is about 240 W.
Friction and air resistance act on the bicycle and the girl when they move. Assume that all the energy is transferred from the battery to the electric motor. Determine the total average resistive force that acts on the bicycle and the girl.
Calculate the component of weight for the bicycle and girl acting down the slope.
The battery continues to give an output power of 240 W. Assume that the resistive forces are the same as in (a)(iii).
Calculate the maximum speed of the bicycle and the girl up the slope.
On another journey up the slope, the girl carries an additional mass. Explain whether carrying this mass will change the maximum distance that the bicycle can travel along the slope.
Determine the internal resistance of the battery.
Calculate the emf of one cell.
Calculate the internal resistance of one cell.
A football player kicks a stationary ball of mass 0.45 kg towards a wall. The initial speed of the ball after the kick is 19 m s−1 and the ball does not rotate. Air resistance is negligible and there is no wind.
The player’s foot is in contact with the ball for 55 ms. Calculate the average force that acts on the ball due to the football player.
The ball leaves the ground at an angle of 22°. The horizontal distance from the initial position of the edge of the ball to the wall is 11 m. Calculate the time taken for the ball to reach the wall.
The top of the wall is 2.4 m above the ground. Deduce whether the ball will hit the wall.
In practice, air resistance affects the ball. Outline the effect that air resistance has on the vertical acceleration of the ball. Take the direction of the acceleration due to gravity to be positive.
The player kicks the ball again. It rolls along the ground without sliding with a horizontal velocity of . The radius of the ball is . Calculate the angular velocity of the ball. State an appropriate SI unit for your answer.
A glider is an aircraft with no engine. To be launched, a glider is uniformly accelerated from rest by a cable pulled by a motor that exerts a horizontal force on the glider throughout the launch.
The glider reaches its launch speed of 27.0 m s–1 after accelerating for 11.0 s. Assume that the glider moves horizontally until it leaves the ground. Calculate the total distance travelled by the glider before it leaves the ground.
The glider and pilot have a total mass of 492 kg. During the acceleration the glider is subject to an average resistive force of 160 N. Determine the average tension in the cable as the glider accelerates.
The cable is pulled by an electric motor. The motor has an overall efficiency of 23 %. Determine the average power input to the motor.
The cable is wound onto a cylinder of diameter 1.2 m. Calculate the angular velocity of the cylinder at the instant when the glider has a speed of 27 m s–1. Include an appropriate unit for your answer.
After takeoff the cable is released and the unpowered glider moves horizontally at constant speed. The wings of the glider provide a lift force. The diagram shows the lift force acting on the glider and the direction of motion of the glider.
Draw the forces acting on the glider to complete the free-body diagram. The dotted lines show the horizontal and vertical directions.
Explain, using appropriate laws of motion, how the forces acting on the glider maintain it in level flight.
At a particular instant in the flight the glider is losing 1.00 m of vertical height for every 6.00 m that it goes forward horizontally. At this instant, the horizontal speed of the glider is 12.5 m s–1. Calculate the velocity of the glider. Give your answer to an appropriate number of significant figures.
A small ball of mass m is moving in a horizontal circle on the inside surface of a frictionless hemispherical bowl.
The normal reaction force N makes an angle θ to the horizontal.
State the direction of the resultant force on the ball.
On the diagram, construct an arrow of the correct length to represent the weight of the ball.
Show that the magnitude of the net force F on the ball is given by the following equation.
The radius of the bowl is 8.0 m and θ = 22°. Determine the speed of the ball.
Outline whether this ball can move on a horizontal circular path of radius equal to the radius of the bowl.
A second identical ball is placed at the bottom of the bowl and the first ball is displaced so that its height from the horizontal is equal to 8.0 m.
The first ball is released and eventually strikes the second ball. The two balls remain in contact. Determine, in m, the maximum height reached by the two balls.
A student strikes a tennis ball that is initially at rest so that it leaves the racquet at a speed of 64 m s–1. The ball has a mass of 0.058 kg and the contact between the ball and the racquet lasts for 25 ms.
The student strikes the tennis ball at point P. The tennis ball is initially directed at an angle of 7.00° to the horizontal.
The following data are available.
Height of P = 2.80 m
Distance of student from net = 11.9 m
Height of net = 0.910 m
Initial speed of tennis ball = 64 m s-1
Calculate the average force exerted by the racquet on the ball.
Calculate the average power delivered to the ball during the impact.
Calculate the time it takes the tennis ball to reach the net.
Show that the tennis ball passes over the net.
Determine the speed of the tennis ball as it strikes the ground.
The student models the bounce of the tennis ball to predict the angle θ at which the ball leaves a surface of clay and a surface of grass.
The model assumes
• during contact with the surface the ball slides.
• the sliding time is the same for both surfaces.
• the sliding frictional force is greater for clay than grass.
• the normal reaction force is the same for both surfaces.
Predict for the student’s model, without calculation, whether θ is greater for a clay surface or for a grass surface.
Cold milk enters a small sterilizing unit and flows over an electrical heating element.
The temperature of the milk is raised from 11 °C to 84 °C. A mass of 55 g of milk enters the sterilizing unit every second.
Specific heat capacity of milk = 3.9 kJ kg−1 K−1
The milk flows out through an insulated metal pipe. The pipe is at a temperature of 84 °C. A small section of the insulation has been removed from around the pipe.
Estimate the power input to the heating element. State an appropriate unit for your answer.
Outline whether your answer to (a) is likely to overestimate or underestimate the power input.
Discuss, with reference to the molecules in the liquid, the difference between milk at 11 °C and milk at 84 °C.
State how energy is transferred from the inside of the metal pipe to the outside of the metal pipe.
The missing section of insulation is 0.56 m long and the external radius of the pipe is 0.067 m. The emissivity of the pipe surface is 0.40. Determine the energy lost every second from the pipe surface. Ignore any absorption of radiation by the pipe surface.
Describe one other method by which significant amounts of energy can be transferred from the pipe to the surroundings.
A lighting system consists of two long metal rods with a potential difference maintained between them. Identical lamps can be connected between the rods as required.
The following data are available for the lamps when at their working temperature.
Lamp specifications 24 V, 5.0 W
Power supply emf 24 V
Power supply maximum current 8.0 A
Length of each rod 12.5 m
Resistivity of rod metal 7.2 × 10–7 Ω m
Each rod is to have a resistance no greater than 0.10 Ω. Calculate, in m, the minimum radius of each rod. Give your answer to an appropriate number of significant figures.
Calculate the maximum number of lamps that can be connected between the rods. Neglect the resistance of the rods.
One advantage of this system is that if one lamp fails then the other lamps in the circuit remain lit. Outline one other electrical advantage of this system compared to one in which the lamps are connected in series.