IB Physics SL A 3 Work Energy and Power Questions

Practise IB Physics A.3 by interpreting force–displacement data, tracing energy transfers and calculating power, efficiency and fuel requirements.

Exam points

Calculate work from force–displacement graphs or W = Fs cos θ, accounting for force direction.
Use resultant work to find a change in kinetic energy, including energy dissipated by friction.
Trace energy transfers between stores and explain conservation, including what a Sankey diagram shows.
Apply mechanical-energy conservation to find speed or height when resistive forces are absent.
Calculate translational kinetic energy with Ek = ½mv² and relate it to momentum using Ek = p²/(2m).
Calculate gravitational or elastic potential-energy changes with mgΔh or ½k(Δx)².
Calculate power from energy transferred per time or from force and velocity using P = Fv.
Use input and useful output energy or power to calculate efficiency and identify dissipated losses.
Use fuel energy density per kilogram or cubic metre to calculate available energy or fuel use.

Covered syllabus points

A.3.1—Conservation of energy A.3.2—Work as energy transfer A.3.3—Sankey diagrams A.3.4—Work by constant force A.3.5—Work-energy change A.3.6—Mechanical energy A.3.7—Mechanical energy conservation A.3.8—Mechanical energy transformations A.3.9—Kinetic energy A.3.10—Gravitational potential energy A.3.11—Elastic potential energy A.3.12—Power A.3.13—Efficiency A.3.14—Fuel energy density

[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

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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)