IB Physics SL A.3 Work, Energy and Power

[Maximum number: 5]

This question is in two parts. Part 1 is about energy resources. Part 2 is about thermal physics.
Part 1 Energy resources
Electricity can be generated using nuclear fission, by burning fossil fuels or using pump storage hydroelectric schemes.

(a)

A hydroelectric scheme has an efficiency of 92 %. Water stored in the dam falls through an average height of 57 m . Determine the rate of flow of water, in $\mathrm{kg} \mathrm{s}^{-1}$ , required to generate an electrical output power of 4.5 MW .

[ 3 ]

use of $\frac{m g h}{t}$ ;
$\frac{m}{t}=\frac{4.5 \times 10^{6}}{0.92 \times 9.81 \times 57}$ ;
$8.7 \times 10^{3} \mathrm{~kg} \mathrm{~s}^{-1}$ ;

(b)

A mass of 0.22 kg of lead spheres is placed in a well-insulated tube. The tube is turned upside down several times so that the spheres fall through an average height of 0.45 m each time the tube is turned. The temperature of the spheres is found to increase by $8^{\circ} \mathrm{C}$ .

[ 2 ]

(i)

Discuss the changes to the energy of the lead spheres.

[ 2 ]

[Maximum number: 11]

This question is in two parts. Part 1 is about the motion of a ship. Part 2 is about melting ice.

(a)

Outline the meaning of work.

[ 2 ]

work done = force × distance moved;
(distance moved) in direction of force;
or
energy transferred;
from one location to another;
or
work done $=F s \cos \theta$ ;
with each symbol defined;

(b)

Some cargo ships use kites working together with the ship's engines to move the vessel.

The tension in the cable that connects the kite to the ship is 250 kN . The kite is pulling the ship at an angle of $39^{\circ}$ to the horizontal. The ship travels at a steady speed of $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ when the ship's engines operate with a power output of 2.7 MW .

[ 6 ]

(i)

Calculate the work done on the ship by the kite when the ship travels a distance of 1.0 km .

[ 2 ]

horizontal force $=250000 \times \cos 39^{\circ}\left(=1.94 \times 10^{5} \mathrm{~N}\right)$ ;
work done $=1.9 \times 10^{8} \mathrm{~J}$ ;

(ii)

Show that, when the ship is travelling at a speed of $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ , the kite provides about 40 % of the total power required by the ship.

[ 4 ]

power provided by kite $=\left(1.94 \times 10^{5} \times 8.5=\right) 1.7 \times 10^{6} \mathrm{~W}$ ;
total power $=(2.7+1.7) \times 10^{6} \mathrm{~W} \quad\left(=4.4 \times 10^{6} \mathrm{~W}\right)$ ;
fraction provided by kite $=\frac{1.7}{2.7+1.7}$ ;
38 % or 0.38 ;
(must see answer to 2+ sig figs as answer is given)

Marking guidance:

Allow answers in the range of 37 to 39 % due to early rounding.
or
Award [3 max] for a reverse argument such as:
if 2.7 MW is 60 %;
then kite power is $\frac{2}{3} \times 2.7 \mathrm{MW}=1.8 \mathrm{MW}$ ;
shows that kite power is actually 1.7 MW ;
(QED)

(c)

The kite is taken down and no longer produces a force on the ship. The resistive force F that opposes the motion of the ship is related to the speed v of the ship by

F=k v^{2}

where k is a constant.
Show that, if the power output of the engines remains at 2.7 MW , the speed of the ship will decrease to about $7 \mathrm{~ms}^{-1}$ . Assume that k is independent of whether the kite is in use or not.

[ 3 ]

$\quad P=\left(k v^{2}\right) \times v=k v^{3}$ ;
$\frac{v_{1}}{v_{2}}=\left(\sqrt[3]{\left(\frac{P_{1}}{P_{2}}\right)}=\right) \sqrt[3]{\left(\frac{2.7}{4.4}\right)} ;$
final speed of ship $=7.2 \mathrm{~ms}^{-1}$ ; (at least 2 sig figs required).

Approximate answer given, marks are for working only.

[Maximum number: 11]

(a)

The conveyor belt moves with a constant horizontal speed of $1.5 \mathrm{~ms}^{-1}$ . As the gravel lands on the belt, it has no horizontal speed.

[ 3 ]

(i)

Calculate the rate of change of the kinetic energy of the gravel due to its change in horizontal speed.

[ 1 ]

$14.6 \mathrm{~J} \mathrm{~s}^{-1}$ ;

(ii)

Determine the power required to move the conveyor belt at constant speed.

[ 2 ]

horizontal momentum gain per second $=13 \times 1.5\left(=19.5 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}\right)$ ;
power required $=29.3 \mathrm{~W}$ ;

(b)

Oscillating water column (OWC) energy converters placed in the ocean are suggested for non-fossil fuel power production.

[ 2 ]

(i)

Outline the energy transformations that take place in an OWC.

[ 2 ]

(c)

(b) (i) An OWC design has an aperture that accepts a wave width of 4.5 m . The waves at the proposed site have an average wavelength of 95 m and wave period of 8.0 s . The overall efficiency of the energy conversion of the OWC is 24 %.

(not to scale)

Assuming that the waves have a rectangular cross-section, determine the minimum wave amplitude that will be required in order for the OWC to produce a power output of 0.10 MW .
Density of water $=1000 \mathrm{~kg} \mathrm{~m}^{-3}$

[ 3 ]

$\frac{P_{\mathrm{required}}}{w}=\frac{0.10\,\mathrm{MW}}{0.24\times4.5\,\mathrm{m}}=0.093\,\mathrm{MW\,m^{-1}}$ $v=\frac{\lambda}{T}=\frac{95\,\mathrm{m}}{8.0\,\mathrm{s}}=11.9\,\mathrm{m\,s^{-1}}$ $\frac{P}{w}=\frac12A^2\rho gv\quad\Longrightarrow\quad A=1.3\,\mathrm{m}$

(d)

On the grid, sketch a labelled Sankey diagram that represents the energy transformation in this OWC.

[ 3 ]

Diagram correct shape (but scaled incorrectly) and labelled.
Waste energy $0.32\ \text{MW}$ (allow ECF from (b)(i)).
All dimensions scaled correctly (by eye).

Figure for Answer to Question (d) — IB Physics SL

[Maximum number: 2]

This question is in two parts. Part 1 is about energy resources. Part 2 is about thermal physics.
Part 1 Energy resources
Electricity can be generated using nuclear fission, by burning fossil fuels or using pump storage hydroelectric schemes.

A mass of 0.22 kg of lead spheres is placed in a well-insulated tube. The tube is turned upside down several times so that the spheres fall through an average height of 0.45 m each time the tube is turned. The temperature of the spheres is found to increase by $8^{\circ} \mathrm{C}$ .

Discuss the changes to the energy of the lead spheres.

[Maximum number: 2]

This question is in two parts. Part 1 is about the motion of a ship. Part 2 is about melting ice.

Outline the meaning of work.

work done = force × distance moved;
(distance moved) in direction of force;
or
energy transferred;
from one location to another;
or
work done $=F s \cos \theta$ ;
with each symbol defined;

[Maximum number: 3]

On the grid, sketch a labelled Sankey diagram that represents the energy transformation in this OWC.

Diagram correct shape (but scaled incorrectly) and labelled.
Waste energy $0.32\ \text{MW}$ (allow ECF from (b)(i)).
All dimensions scaled correctly (by eye).

Figure for Answer to Question 6 — IB Physics SL

[Maximum number: 2]

This question is in two parts. Part 1 is about two children on a merry-go-round. Part 2 is about electric circuits.

Part 1 Two children on a merry-go-round
Aibhe and Euan are sitting on opposite sides of a merry-go-round, which is rotating at constant speed around a fixed centre. The diagram below shows the view from above.

Aibhe is moving at speed $1.0 \mathrm{~ms}^{-1}$ relative to the ground.

Euan drags his foot along the ground to act as a brake. The merry-go-round comes to a stop after 4.0 rotations. The radius of the merry-go-round is 1.5 m . The average frictional force between his foot and the ground is 45 N . Calculate the work done.

distance travelled by Euan $=4.0 \times 2 \pi \times 1.5(=37.70 \mathrm{~m})$ ;

W\left(=F_{a v} d=45 \times 37.70\right)=1700(\mathrm{~J}) ;

[Maximum number: 4]

This question is about the motion of a bicycle.

A cyclist is moving up a slope that is at an angle of $19^{\circ}$ to the horizontal. The mass of the cyclist and the bicycle is 85 kg .

At the bottom of the slope the cyclist has a speed of $5.5 \mathrm{~m} \mathrm{~s}^{-1}$ . The cyclist stops pedalling and applies the brakes which provide an additional decelerating force of 250 N . Determine the distance taken for the cyclist to stop. Assume air resistance is negligible and that there are no other frictional forces.

total decelerating force $=271+250(=521 \mathrm{~N})$ ;
acceleration $=(-) \frac{521}{85}\left(=-6.13 \mathrm{~ms}^{-2}\right)$ ;
$s=\frac{v^{2}-u^{2}}{2 a} ;$ $2.47(\mathrm{~m}) ;\} \begin{aligned} & \text { (signs must be consistent for this mark, ie: if acceleration assumed } \\ & \text { positive, look for negative distance) }\end{aligned}$

Allow use of g=10. Answers are $527 \mathrm{~N}, 6.2 \mathrm{~m} \mathrm{~s}^{-2}, 2.44 \mathrm{~m}$ .
or
total decelerating force $=271+250(=521 \mathrm{~N})$ ;
initial kinetic energy $=\frac{1}{2} m v^{2}=1290 \mathrm{~J}$ ;
distance $=\frac{\text { energy lost }}{\text { force }}=\frac{1290}{521}$ ;
2.47 (m);

[Maximum number: 2]

A student strikes a tennis ball that is initially at rest so that it leaves the racquet at a speed of $64 \mathrm{~m} \mathrm{~s}^{-1}$ . 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 $7.00^{\circ}$ to the horizontal.

The following data are available.

Determine the speed of the tennis ball as it strikes the ground.

ALTERNATIVE 1

Initial KE+PE= final KE /
$\frac{1}{2} \times 0.058 \times 64^{2}+0.058 \times 9.81 \times 2.80=\frac{1}{2} \times 0.058 \times v^{2}$

ALTERNATIVE 2
$\begin{aligned} & v_{v}=<\sqrt{7.8^{2}+2 \times 9.81 \times 2.8}>=10.8<\mathrm{ms}^{-1} \gg & <v=\sqrt{63.5^{2}+10.8^{2}}> & v=64.4<\mathrm{ms}^{-1} \gg \end{aligned}$

[Maximum number: 2]

An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.

The unextended length of the rope is 60.0 m . From position A to position B, the block falls freely.

(a)

For the rope and block, describe the energy changes that take place

[ 2 ]

(i)

between A and B .

[ 1 ]

(ii)

between B and C .

[ 1 ]

[Maximum number: 2]

A toy rocket is made from a plastic bottle that contains some water.

Air is pumped into the vertical bottle until the pressure inside forces water and air out of the bottle. The bottle then travels vertically upwards.

The air-water mixture is called the propellant.
The variation with time of the vertical velocity of the bottle is shown.

The bottle reaches its highest point at time $T_{1}$ on the graph and returns to the ground at time $T_{2}$ . The bottle then bounces. The motion of the bottle after the bounce is shown as a dashed line.

The bottle bounces when it returns to the ground.

Calculate the fraction of the kinetic energy of the bottle that remains after the bounce.

Attempt to evaluate KE ratio as $\left(\frac{v_{\text {final }}}{v_{\text {initial }}}\right)^{2}$

« $\left(\frac{4.5}{10}\right)^{2}=$ 》 0.20 OR 20 % OR $\frac{1}{5}$

Marking guidance:

Accept ± 0.5 velocity values from graph

[Maximum number: 3]

This question is about a tidal power station.

A tidal power station is built for a coastal town. Sea water is stored in a tidal basin behind a dam at high tide and released in a controlled manner between high tides, so that it passes through turbines to generate electricity.

The following data are available.

Calculate the electrical energy produced between successive high and low tides.

difference in height of centre of mass of water $=\frac{1.8}{2}=0.9(\mathrm{~m})$ ;
$\Delta E_{\mathrm{P}}\left(=m g \Delta h=2.8 \times 10^{8} \times 9.8 \times 0.9\right)=2.5 \times 10^{9}(\mathrm{~J}) ;$
electrical energy $\left(=0.24 \times 2.5 \times 10^{9}\right)=5.9 \times 10^{8}(\mathrm{~J})(=590 \mathrm{MJ})$ ;

Marking guidance:

Allow ECF for [2 max] if candidate omits factor of 2 in first marking points.
Accept $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ giving an answer of $6.0 \times 10^{8}(\mathrm{~J})$ .

[Maximum number: 2]

An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.

The unextended length of the rope is 60.0 m . From position A to position B, the block falls freely.

The length reached by the rope at C is 77.4 m . Suggest how energy considerations could be used to determine the elastic constant of the rope.

k can be determined using EPE $=\frac{1}{2} k x^{2}$

correct statement or equation showing

GPE at A=EPE at C

OR

( GPE+KE ) at B=EPE at C

Candidate must clearly indicate the energy associated with either position A or B for MP2.

[Maximum number: 2]

A 300 W immersion heater is placed in a beaker containing 0.25 kg of water at a temperature of $18^{\circ} \mathrm{C}$ . The heater is switched on for 120 s , after which time the temperature of the water is $45^{\circ} \mathrm{C}$ . The thermal capacity of the beaker is negligible and the specific heat capacity of water is $4.2 \times 10^{3} \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$ .

Determine the rate at which thermal energy is transferred from the water to the surroundings during the time that the heater is switched on.

energy transfer $=[300 \times 120]-\left[2.835 \times 10^{4}\right]=7.65 \times 10^{3} \mathrm{~J}$ ;
rate of transfer $=\frac{7.650 \times 10^{3}}{120}=64 \mathrm{~W}$ ;

Marking guidance:

Allow ECF from (b)(i).

[Maximum number: 3]

This question is in two parts. Part 1 is about renewable energy. Part 2 is about nuclear energy and radioactivity.

A small coastal community decides to use a wind farm consisting of five identical wind turbines to generate part of its energy. At the proposed site, the average wind speed is $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ and the density of air is $1.3 \mathrm{~kg} \mathrm{~m}^{-3}$ . The maximum power required from the wind farm is 0.75 MW . Each turbine has an efficiency of 30 %.

Determine the diameter that will be required for the turbine blades to achieve the maximum power of 0.75 MW .

total wind power required $=\frac{750000}{0.3}$ ;

Marking guidance:

maximum wind power required per turbine, $P=\frac{750000}{5 \times 0.3}(=500 \mathrm{~kW})$ ;
$d=\left(\frac{8 P}{\rho \pi v^{3}}=\right)^{\frac{1}{2}} 40(\mathrm{~m}) ;$
[3]
Award [1 max] for an answer of 48.9(m) as it indicates 5 and 0.3 ignored.
Award [2 max] for 22(m) as it indicates 0.3 ignored.
Award [2 max] for 89(m) as it indicates 5 ignored.

[Maximum number: 2]

This question is in two parts. Part 1 is about renewable energy. Part 2 is about nuclear energy and radioactivity.

A small coastal community decides to use a wind farm consisting of five identical wind turbines to generate part of its energy. At the proposed site, the average wind speed is $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ and the density of air is $1.3 \mathrm{~kg} \mathrm{~m}^{-3}$ . The maximum power required from the wind farm is 0.75 MW . Each turbine has an efficiency of 30 %.

Currently, a nearby coal-fired power station generates energy for the community. Less coal will be burnt at the power station if the wind farm is constructed.

The energy density of coal is $35 \mathrm{MJ} \mathrm{kg}^{-1}$ . Estimate the minimum mass of coal that can be saved every hour when the wind farm is producing its full output.

mass of coal per second $(=0.0214 \mathrm{~kg})$ ;
$77.1(\mathrm{~kg})$ ;
or
energy saved per hour $=0.75 \times 3600\left(=2700 \mathrm{MJ} \mathrm{h}^{-1}\right)$ ;
mass of coal saved $=\left(\frac{2700}{35}=\right) 77.1(\mathrm{~kg})$ ;

Marking guidance:

Award [2] for a bald correct answer.

IB Physics SL A 3 Work Energy and Power Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (i)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question 3

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)

Question (c)

Question (d)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question (a)

Question (i)

Question (ii)

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16