ENGAA 2021

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

Two loudspeakers are positioned 8.0 m apart as shown.

The loudspeakers emit sound waves of the same single frequency. The wave emitted by one loudspeaker is 180° out of phase with the wave emitted by the other loudspeaker.

A point P is in front of the loudspeakers. P is 18.0 m from one loudspeaker and 24.0 m from the other loudspeaker. As a result of superposition of the two waves arriving at P, the amplitude of the sound at position P is a minimum.

The speed of the sound is 336 ms⁻¹.

What is the lowest possible frequency of the sound?

• A.21 Hz
• B.28 Hz
• C.42 Hz
• D.56 Hz
• E.63 Hz
• F.84 Hz

Answer: D

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.$F < N < W$
• B.$F < W < N$
• C.$N < F < W$
• D.$N < W < F$
• E.$W < F < N$
• F.$W < N < F$

Answer: C

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.4.0 V
• B.6.0 V
• C.9.9 V
• D.10.0 V
• E.10.1 V
• F.12.0 V
• G.14.0 V

Answer: G

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

Two objects of mass M and m are connected by a rope over a pulley on an inclined plane as shown.

[diagram not to scale]

There is no friction between the plane and the object. The pulley is smooth, and the rope has negligible mass.

The angle $\theta$ of the plane to the horizontal is such that $\sin \theta = 0.80$ and $\cos \theta = 0.60$.

The object with mass M accelerates down the slope.

Which expression describes the full range of possible values of M compared with m?

• A.$M > \frac{3}{5}m$
• B.$M > \frac{4}{5}m$
• C.$M > m$
• D.$M > \frac{5}{4}m$
• E.$M > \frac{5}{3}m$

Answer: D

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

An object P falls vertically from rest through air and reaches terminal velocity.

An identical object Q is projected vertically upwards from the ground.

When Q reaches its maximum height, P collides with it. The two objects join together in such a way that there is no change to the area of cross section passing through the air.

The two combined objects then fall through the air as one object.

Which sketch graph shows the variation of velocity with time for object P before and after the collision?

• A.Graph A
• B.Graph B
• C.Graph C
• D.Graph D
• E.Graph E
• F.Graph F

Answer: F

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.$\frac{1}{3}$
• B.$\frac{1}{2}$
• C.1
• D.2
• E.3

Answer: A

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.2.7 W
• B.3.6 W
• C.7.5 W
• D.10.8 W
• E.13.5 W
• F.43.2 W

Answer: D

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

A light spring is used to support a uniform rod horizontally against a wall as shown. The angle between the spring and the rod is $\theta$.

The spring constant of the spring is 20 Nm⁻¹ and the weight of the rod is 16 N.

The angle $\theta$ is such that $\cos \theta = \frac{3}{5}$ and $\sin \theta = \frac{4}{5}$.

How much energy is stored in the spring?

• A.1.6 J
• B.2.5 J
• C.3.2 J
• D.4.4 J
• E.5.0 J
• F.6.4 J
• G.10 J
• H.40 J

Answer: B

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.6.7 kg ms⁻²
• B.66.7 kg ms⁻²
• C.120 kg ms⁻²
• D.240 kg ms⁻²
• E.600 kg ms⁻²

Answer: D

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

In an industrial process to test the purity of a metal, a narrow beam of ultrasound passes into a block of the metal. The ultrasound generator U is immersed in a gel that is in contact with the metal. The ultrasound passes from the gel into the metal.

The arcs of circles shown in the gel are lines that represent the positions of the compressions (known as wavefronts) of the ultrasound wave that comes from U.

Ultrasound travels faster in the metal than in the gel.

The wavefronts in the metal are circular arcs with their centre at a point X that is on the dashed line.

Where on the dashed line is X?

• A.above U
• B.at U
• C.in the gel below U
• D.on the boundary between the gel and the metal
• E.in the metal

Answer: C

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

The diagram shows a circuit containing two power supplies with negligible internal resistance and two resistors with resistances R and 5R.

The emfs of the power supplies and the magnitude and direction of the current in one part of the circuit are shown.

One point in the circuit is labelled P.

What is the magnitude of the current at P?

• A.3.0 mA
• B.7.0 mA
• C.8.5 mA
• D.11.5 mA
• E.13 mA
• F.25 mA

Answer: E

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

A selection of five wires made from the same metal have different unstretched lengths but equal masses. The wires are all subjected to the same small tension force and each wire extends within its limit of proportionality.

Which graph shows the relationship between the extension of the wires and the unstretched length of the wires?

• A.Graph A: extension is constant with unstretched length.
• B.Graph B: extension is directly proportional to unstretched length.
• C.Graph C: extension is inversely proportional to unstretched length.
• D.Graph D: extension increases with unstretched length, with a decreasing slope.
• E.Graph E: extension decreases with unstretched length, hyperbolically.
• F.Graph F: extension increases quadratically with unstretched length.

Answer: F

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

Water enters a horizontal pipe of cross-sectional area 0.0040 m² at constant speed 0.50 ms⁻¹. At the end of the pipe the cross-sectional area reduces to 0.0020 m² and the water leaves the pipe as shown. The density of water is 1000 kg m⁻³.

How much power must be supplied to the water to maintain the flow in this section of the pipe?

(Assume that the water is incompressible and that frictional forces can be neglected.)

• A.0.25 W
• B.0.50 W
• C.0.75 W
• D.1.0 W
• E.1.25 W
• F.1.5 W
• G.3.75 W

Answer: C

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

Two light wires P and Q support a load of weight W in equilibrium as shown. Wire P is horizontal and wire Q is at an angle of 60° to the vertical. The wires are made from the same material.

The radius of wire Q is twice the radius of wire P.

What is the ratio

$$\frac{\text{strain\;in\;wire\;P}}{\text{strain\;in\;wire\;Q}} ?$$

(The wires do not exceed their limits of proportionality.)

• A.$\frac{\sqrt{3}}{8}$
• B.$\frac{\sqrt{3}}{4}$
• C.$\frac{\sqrt{3}}{2}$
• D.$\sqrt{3}$
• E.$2\sqrt{3}$
• F.$\frac{4}{\sqrt{3}}$
• G.$\frac{8}{\sqrt{3}}$

Answer: E

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

The speed of light in a block of glass is $2.0 \times 10^8 \text{ms}^{-1}$. The block of glass is immersed in a liquid of refractive index 1.2.

The diagram shows a ray of light travelling in the glass block striking the side of the block at the point labelled X. The acute angle between the ray and the side of the block is $\theta$.

What is the full range of values of the acute angle $\theta$ for which light is refracted at X?

(The speed of light in a vacuum is $3.0 \times 10^8 \text{ms}^{-1}$.)

• A.$0° < \theta < \cos^{-1}(\frac{2}{3})$
• B.$0° < \theta < \cos^{-1}(\frac{\sqrt{5}}{3})$
• C.$0° < \theta < \cos^{-1}(\frac{3}{5})$
• D.$0° < \theta < \cos^{-1}(\frac{4}{5})$
• E.$\cos^{-1}(\frac{2}{3}) < \theta < 90°$
• F.$\cos^{-1}(\frac{\sqrt{5}}{3}) < \theta < 90°$
• G.$\cos^{-1}(\frac{3}{5}) < \theta < 90°$
• H.$\cos^{-1}(\frac{4}{5}) < \theta < 90°$

Answer: H

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

A car is at rest on a straight horizontal road. At time t = 0 s the car starts to move along the road. The graph shows how its acceleration varies from t = 0 s to t = 20 s.

What is the displacement of the car from its starting position when t = 20 s?

• A.5.0 m
• B.25 m
• C.35 m
• D.175 m
• E.225 m
• F.375 m

Answer: E

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

An empty measuring cylinder is placed on a balance, and the balance reading is then set to zero.

A mass of 8.7 g of a powder is poured into the measuring cylinder as shown in the diagram.

Liquid is poured into the cylinder to cover the powder completely. The powder does not dissolve. The reading on the measuring cylinder and the reading on the balance are recorded.

More liquid is added and a second pair of readings is recorded.

The table shows the two pairs of readings.

| reading on measuring cylinder / cm³ | reading on balance / g |
| :--- | :--- |
| 10.0 | 15.0 |
| 25.0 | 27.6 |

What is the density of the material from which the powder is made?

• A.0.414 g cm⁻³
• B.1.16 g cm⁻³
• C.1.31 g cm⁻³
• D.1.45 g cm⁻³
• E.2.00 g cm⁻³
• F.2.50 g cm⁻³
• G.3.48 g cm⁻³
• H.6.00 g cm⁻³

Answer: G

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.60 m
• B.80 m
• C.120 m
• D.160 m
• E.200 m

Answer: C

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

• A.$\frac{Vk}{3} \left( \frac{I_F}{k} \right)^{\frac{3}{2}}$
• B.$Vk \left( \frac{I_F}{k} \right)^{\frac{3}{2}}$
• C.$3Vk \left( \frac{I_F}{k} \right)^{\frac{3}{2}}$
• D.$\frac{Vk}{3} \left( \frac{I_F}{k} \right)$
• E.$Vk \left( \frac{I_F}{k} \right)$
• F.$3Vk \left( \frac{I_F}{k} \right)$

Answer: A

Not in the current ESAT specification — included for completeness, but you won't be tested on it.

A water trough has the shape of a prism, with a cross section that is a right-angled isosceles triangle.

One rectangular face and the two triangular ends of the trough are vertical, as shown.

[diagram not to scale]

The trough contains water of depth 0.60 m measured on the vertical rectangular face.

What is the force exerted by the water on one triangular end of the trough?

(density of water = 1000 kg m⁻³; gravitational field strength = 10 N kg⁻¹)

• A.180 N
• B.270 N
• C.360 N
• D.540 N
• E.720 N
• F.1080 N
• G.6000 N
• H.12000 N

Answer: C