HL Paper 3
A mass-spring system is forced to vibrate vertically at the resonant frequency of the system. The motion of the system is damped using a liquid.
At time t=0 the vibrator is switched on. At time tB the vibrator is switched off and the system comes to rest. The graph shows the variation of the vertical displacement of the system with time until tB.
Explain, with reference to energy in the system, the amplitude of oscillation between
(i) t = 0 and tA.
(ii) tA and tB.
The system is critically damped. Draw, on the graph, the variation of the displacement with time from tB until the system comes to rest.
A ball is moving in still air, spinning clockwise about a horizontal axis through its centre. The diagram shows streamlines around the ball.
The surface area of the ball is 2.50 x 10–2 m2. The speed of air is 28.4 ms–1 under the ball and 16.6 ms–1 above the ball. The density of air is 1.20 kgm–3.
Estimate the magnitude of the force on the ball, ignoring gravity.
On the diagram, draw an arrow to indicate the direction of this force.
State one assumption you made in your estimate in (a)(i).
The graph below shows the displacement y of an oscillating system as a function of time t.
State what is meant by damping.
Calculate the Q factor for the system.
The Q factor of the system increases. State and explain the change to the graph.
A solid cube of side 0.15 m has an average density of 210 kg m–3.
(i) Calculate the weight of the cube.
(ii) The cube is placed in gasoline of density 720 kg m–3. Calculate the proportion of the volume of the cube that is above the surface of the gasoline.
Water flows through a constricted pipe. Vertical tubes A and B, open to the air, are located along the pipe.
Describe why tube B has a lower water level than tube A.
An air bubble has a radius of 0.25 mm and is travelling upwards at its terminal speed in a liquid of viscosity 1.0 × 10–3 Pa s.
The density of air is 1.2 kg m–3 and the density of the liquid is 1200 kg m–3.
Explain the origin of the buoyancy force on the air bubble.
With reference to the ratio of weight to buoyancy force, show that the weight of the air bubble can be neglected in this situation.
Calculate the terminal speed.
A sphere is dropped into a container of oil.
The following data are available.
Density of oil
Viscosity of oil
Volume of sphere
Mass of sphere
The sphere is now suspended from a spring so that the sphere is below the surface of the oil.
State two properties of an ideal fluid.
Determine the terminal velocity of the sphere.
Determine the force exerted by the spring on the sphere when the sphere is at rest.
The sphere oscillates vertically within the oil at the natural frequency of the sphere-spring system. The energy is reduced in each cycle by . Calculate the factor for this system.
Outline the effect on of changing the oil to one with greater viscosity.
The graph below represents the variation with time t of the horizontal displacement x of a mass attached to a vertical spring.
The total mass for the oscillating system is 30 kg. For this system
Describe the motion of the spring-mass system.
determine the initial energy.
calculate the Q at the start of the motion.
Two tubes, A and B, are inserted into a fluid flowing through a horizontal pipe of diameter 0.50 m. The openings X and Y of the tubes are at the exact centre of the pipe. The liquid rises to a height of 0.10 m in tube A and 0.32 m in tube B. The density of the fluid = 1.0 × 103 kg m–3.
The viscosity of water is 8.9 × 10–4 Pa s.
Show that the velocity of the fluid at X is about 2 ms–1, assuming that the flow is laminar.
Estimate the Reynolds number for the fluid in your answer to (a).
Outline whether your answer to (a) is valid.
A driven system is lightly damped. The graph shows the variation with driving frequency f of the amplitude A of oscillation.
A mass on a spring is forced to oscillate by connecting it to a sine wave vibrator. The graph shows the variation with time t of the resulting displacement y of the mass. The sine wave vibrator has the same frequency as the natural frequency of the spring–mass system.
On the graph, sketch a curve to show the variation with driving frequency of the amplitude when the damping of the system increases.
State and explain the displacement of the sine wave vibrator at t = 8.0 s.
The vibrator is switched off and the spring continues to oscillate. The Q factor is 25.
Calculate the ratio for the oscillations of the spring–mass system.
The diagram shows a simplified model of a Galilean thermometer. The thermometer consists of a sealed glass cylinder that contains ethanol, together with glass spheres. The spheres are filled with different volumes of coloured water. The mass of the glass can be neglected as well as any expansion of the glass through the temperature range experienced. Spheres have tags to identify the temperature. The mass of the tags can be neglected in all calculations.
Each sphere has a radius of 3.0 cm and the spheres, due to the different volumes of water in them, are of varying densities. As the temperature of the ethanol changes the individual spheres rise or fall, depending on their densities, compared with that of the ethanol.
The graph shows the variation with temperature of the density of ethanol.
Using the graph, determine the buoyancy force acting on a sphere when the ethanol is at a temperature of 25 °C.
When the ethanol is at a temperature of 25 °C, the 25 °C sphere is just at equilibrium. This sphere contains water of density 1080 kg m–3. Calculate the percentage of the sphere volume filled by water.
The room temperature slightly increases from 25 °C, causing the buoyancy force to decrease. For this change in temperature, the ethanol density decreases from 785.20 kg m–3 to 785.16 kg m–3. The average viscosity of ethanol over the temperature range covered by the thermometer is 0.0011 Pa s. Estimate the steady velocity at which the 25 °C sphere falls.
The water supply for a hydroelectric plant is a reservoir with a large surface area. An outlet pipe takes the water to a turbine.
The following data are available:
State the difference in terms of the velocity of the water between laminar and turbulent flow.
The water level is a height H above the turbine. Assume that the flow is laminar in the outlet pipe.
Show, using the Bernouilli equation, that the speed of the water as it enters the turbine is given by v = .
Calculate the Reynolds number for the water flow.
Outline whether it is reasonable to assume that flow is laminar in this situation.
A railway track passes over a bridge that has a span of 20 m.
The bridge is subject to a periodic force as a train crosses, this is caused by the weight of the train acting through the wheels as they pass the centre of the bridge.
The wheels of the train are separated by 25 m.
The graph shows the variation of the amplitude of vibration A of the bridge with driving frequency fD, when the damping of the bridge system is small.
Show that, when the speed of the train is 10 m s-1, the frequency of the periodic force is 0.4 Hz.
Outline, with reference to the curve, why it is unsafe to drive a train across the bridge at 30 m s-1 for this amount of damping.
The damping of the bridge system can be varied. Draw, on the graph, a second curve when the damping is larger.
A mass is attached to a vertical spring. The other end of the spring is attached to the driver of an oscillator.
The mass is performing very lightly damped harmonic oscillations. The frequency of the driver is higher than the natural frequency of the system. At one instant the driver is moving downwards.
State and explain the direction of motion of the mass at this instant.
The oscillator is switched off. The system has a Q factor of 22. The initial amplitude is 10 cm. Determine the amplitude after one complete period of oscillation.
A pendulum bob is displaced until its centre is 30 mm above its rest position and then released. The motion of the pendulum is lightly damped.
Describe what is meant by damped motion.
After one complete oscillation, the height of the pendulum bob above the rest position has decreased to 28 mm. Calculate the Q factor.
The point of suspension now vibrates horizontally with small amplitude and frequency 0.80 Hz, which is the natural frequency of the pendulum. The amount of damping is unchanged.
When the pendulum oscillates with a constant amplitude the energy stored in the system is 20 mJ. Calculate the average power, in W, delivered to the pendulum by the driving force.
Gasoline of density 720 kg m–3 flows in a pipe of constant diameter.
State one condition that must be satisfied for the Bernoulli equation
ρv2 + ρgz + ρ = constant
to apply
Outline why the speed of the gasoline at X is the same as that at Y.
Calculate the difference in pressure between X and Y.
The diameter at Y is made smaller than that at X. Explain why the pressure difference between X and Y will increase.
A solid sphere is released from rest below the surface of a fluid and begins to fall.
Draw and label the forces acting on the sphere at the instant when it is released.
Explain why the sphere will reach a terminal speed.
The weight of the sphere is 6.16 mN and the radius is 5.00 × 10-3 m. For a fluid of density 8.50 × 102 kg m-3, the terminal speed is found to be 0.280 m s-1. Calculate the viscosity of the fluid.
The graph shows the variation with time t of the total energy E of a damped oscillating system.
The Q factor for the system is 25. Determine the period of oscillation for this system.
Another system has the same initial total energy and period as that in (a) but its Q factor is greater than 25. Without any calculations, draw on the graph, the variation with time of the total energy of this system.
A horizontal pipe is inserted into the cylindrical tube so that its centre is at a depth of 5.0 m from the surface of the water. The diameter D of the pipe is half that of the tube.
When the pipe is opened, water exits the pipe with speed u and the surface of the water in the tube moves downwards with speed v.
An ice cube floats in water that is contained in a tube.
The ice cube melts.
Suggest what happens to the level of the water in the tube.
Outline why u = 4v.
The density of water is 1000 kg m–3. Calculate u.
A farmer is driving a vehicle across an uneven field in which there are undulations every 3.0 m.
The farmer’s seat is mounted on a spring. The system, consisting of the mass of the farmer and the spring, has a natural frequency of vibration of 1.9 Hz.
Explain why it would be uncomfortable for the farmer to drive the vehicle at a speed of 5.6 m s–1.
Outline what change would be required to the value of Q for the mass–spring system in order for the drive to be more comfortable.
A Pitot tube shown in the diagram is used to determine the speed of air flowing steadily in a horizontal wind tunnel. The narrow tube between points A and B is filled with a liquid. At point B the speed of the air is zero.
Explain why the levels of the liquid are at different heights.
The density of the liquid in the tube is 8.7 × 102 kg m–3 and the density of air is 1.2 kg m–3. The difference in the level of the liquid is 6.0 cm. Determine the speed of air at A.
The natural frequency of a driven oscillating system is 6 kHz. The frequency of the driver for the system is varied from zero to 20 kHz.
Draw a graph to show the variation of amplitude of oscillation of the system with frequency.
The Q factor for the system is reduced significantly. Describe how the graph you drew in (a) changes.