ENGAA 2022 D564/12

The diagram shows an object of mass 2.4 kg on a smooth horizontal surface.

A force $F$ acts on the object at an acute angle $\theta$ to the horizontal, where $\tan \theta = \frac{4}{3}$.

A force of 36 N acts on the object towards the right.

The object is in equilibrium.

What is the magnitude of the normal contact force exerted on the object by the surface?

(gravitational field strength = $10 \text{Nkg}^{-1}$)

• A.24 N
• B.27 N
• C.48 N
• D.51 N
• E.72 N
• F.75 N

Answer: E

The length of a spring when no force acts on it is $L$. The spring constant of the spring is $3.0 \times 10^3 \text{ Nm}^{-1}$.

The spring is on the floor of an accelerating lift (elevator), and the spring supports a 30 kg mass.

The lift is accelerating downwards at $2.0 \text{ ms}^{-2}$.

What is the difference between $L$ and the length of the spring when the lift is accelerating downwards?

(gravitational field strength = $10 \text{ Nkg}^{-1}$; the spring obeys Hooke's law)

• A.0 cm
• B.2.0 cm
• C.8.0 cm
• D.10 cm
• E.12 cm

Answer: C

• A.1.28 kW
• B.2.56 kW
• C.5.12 kW
• D.32 kW
• E.64 kW
• F.128 kW

Answer: C

• A.$1.8 \text{ ms}^{-1}$
• B.$3.6 \text{ ms}^{-1}$
• C.$7.2 \text{ ms}^{-1}$
• D.$62.5 \text{ ms}^{-1}$
• E.$125 \text{ ms}^{-1}$
• F.$250 \text{ ms}^{-1}$

Answer: E

Three light springs, P, Q and R, are identical.

Springs P and Q are connected in series as shown. A downwards force $T$ is applied to the lower end.

Spring R is cut into four equal lengths, and the four pieces arranged symmetrically as shown. The two connecting bars have negligible mass. A downwards force $F$ is applied to the centre of the lower bar.

The total extensions of the two systems are equal. The springs obey Hooke's law.

Which expression gives $T$ in terms of $F$?

• A.$\frac{F}{16}$
• B.$\frac{F}{8}$
• C.$\frac{F}{4}$
• D.$\frac{F}{2}$
• E.$2F$
• F.$4F$
• G.$8F$
• H.$16F$

Answer: B

A nylon cube resting on a horizontal surface has a volume of $64 \text{ cm}^3$.

A force $F$ is applied vertically downwards on the top face of the cube so that it compresses the height by $x$.

The graph shows the variation of $F$ with $x$.

What is the Young modulus of the nylon?

(Assume that changes in horizontal cross-sectional area are negligible.)

• A.$7.7 \times 10^3 \text{ Pa}$
• B.$4.8 \times 10^6 \text{ Pa}$
• C.$9.6 \times 10^6 \text{ Pa}$
• D.$1.2 \times 10^8 \text{ Pa}$
• E.$1.5 \times 10^9 \text{ Pa}$
• F.$3.0 \times 10^9 \text{ Pa}$
• G.$1.9 \times 10^{12} \text{ Pa}$

Answer: F

• A.1.5 s
• B.2.0 s
• C.2.5 s
• D.3.0 s
• E.3.5 s
• F.4.0 s
• G.5.0 s

Answer: D

• A.0.60 J
• B.0.70 J
• C.1.9 J
• D.2.9 J
• E.4.5 J
• F.17 J
• G.26 J
• H.41 J

Answer: D

• A.$\sin^{-1} \frac{\sqrt{3}}{4}$
• B.$\sin^{-1} \frac{1}{\sqrt{3}}$
• C.$\sin^{-1} \frac{2}{\sqrt{6}}$
• D.$\sin^{-1} \frac{\sqrt{3}}{2}$
• E.$\sin^{-1} 1$

Answer: B

• A.$2.8 \text{kgms}^{-1}$
• B.$9.0 \text{kgms}^{-1}$
• C.$55 \text{kgms}^{-1}$
• D.$80 \text{kgms}^{-1}$
• E.$100 \text{kgms}^{-1}$
• F.$180 \text{kgms}^{-1}$

Answer: F

Two small hard spheres of mass $m$ and $2m$ are suspended side by side from light vertical strings of length $l$. The more massive sphere is raised so that its string is horizontal, and then released. It swings through 90° and strikes the smaller sphere. The two spheres stick together, and rise to a maximum height $h$ as shown in the diagram.

[diagram not to scale]

Which expression gives the height $h$ in terms of $l$?

(Assume that air resistance is negligible.)

• A.$\frac{4l}{27}$
• B.$\frac{8l}{27}$
• C.$\frac{4l}{9}$
• D.$\frac{2l}{3}$
• E.$\frac{8l}{9}$
• F.$l$
• G.$2l$

Answer: C

Three resistance wires X, Y and Z, made from the same metal, are connected to each other and to a circular plastic ring as shown.

[diagram not to scale]

Wires X and Y each have twice the diameter of wire Z.

Wire X is 12 cm long. Wire Z is 15 cm long and is connected across a diameter of the ring.

A power supply is connected to the two corners of the triangle that lie on the diameter.

What is the value of the ratio $\frac{\text{current\;in\;X}}{\text{current\;in\;Z}}$?

• A.$\frac{1}{5}$
• B.$\frac{7}{20}$
• C.$\frac{7}{10}$
• D.$\frac{5}{7}$
• E.$\frac{7}{5}$
• F.$\frac{10}{7}$
• G.$\frac{20}{7}$
• H.5

Answer: G

• A.$5.2 \times 10^8 \text{ Nm}^{-2}$
• B.$8.0 \times 10^8 \text{ Nm}^{-2}$
• C.$1.0 \times 10^9 \text{ Nm}^{-2}$
• D.$1.3 \times 10^9 \text{ Nm}^{-2}$
• E.$2.0 \times 10^9 \text{ Nm}^{-2}$
• F.$2.6 \times 10^9 \text{ Nm}^{-2}$
• G.$5.2 \times 10^9 \text{ Nm}^{-2}$

Answer: F

The three graphs show the displacement, velocity and acceleration against time for an object moving in a straight line.

The time axis is shown to the same scale on all three graphs.

Which graph represents which quantity?

• A.graph P: acceleration, graph Q: displacement, graph R: velocity
• B.graph P: acceleration, graph Q: velocity, graph R: displacement
• C.graph P: displacement, graph Q: acceleration, graph R: velocity
• D.graph P: displacement, graph Q: velocity, graph R: acceleration
• E.graph P: velocity, graph Q: acceleration, graph R: displacement
• F.graph P: velocity, graph Q: displacement, graph R: acceleration

Answer: C

• A.1.2 kJ
• B.60 kJ
• C.70 kJ
• D.120 kJ
• E.140 kJ
• F.740 kJ

Answer: E

The diagram shows a circuit that includes two batteries, each with negligible internal resistance.

What is the reading on the ammeter?

• A.0.0029 A
• B.0.0071 A
• C.0.063 A
• D.0.083 A
• E.0.50 A
• F.0.65 A
• G.1.2 A
• H.2.0 A

Answer: E

The upper diagram shows the equilibrium positions of nine equally spaced particles in a medium.

The lower diagram shows the positions of the same nine particles when a longitudinal wave is travelling through the medium. The wave is shown at time $t = 0$, travelling to the right.

The frequency of the wave is 0.5 Hz.

Which graph represents the displacements of the particles at a later time $t = 0.5$s?

(On the graphs, positive displacement values represent particle displacements to the right.)

• A.Graph A: a sine wave.
• B.Graph B: a negative sine wave.
• C.Graph C: a cosine wave.
• D.Graph D: a negative cosine wave.
• E.Graph E: a sine wave with half the wavelength of A.
• F.Graph F: a negative sine wave with half the wavelength of B.
• G.Graph G: a cosine wave with half the wavelength of C.
• H.Graph H: a cosine wave with half the wavelength of D.

Answer: A

A power supply with constant emf and internal resistance $r$ is connected to an external resistor.

The efficiency of the system is defined as
$\text{efficiency} = \frac{\text{power\;dissipated\;by\;external\;resistor}}{\text{total\;power\;supplied\;by\;cell}}$

Which graph shows how the efficiency varies with the resistance of the external resistor?

• A.Graph A: curve starting at (0,0), passing through (r, 0.5) and asymptotically approaching 1.
• B.Graph B: curve starting at (0,0), peaking, then decreasing towards 0.
• C.Graph C: horizontal line at efficiency = 0.5 for resistance > 0.
• D.Graph D: curve starting at (0,0) and asymptotically approaching 0.5.
• E.Graph E: curve starting at (0,0), peaking at resistance r, then decreasing towards 0.
• F.Graph F: curve starting at (0,1) and decreasing towards 0.

Answer: A

• A.$2 \text{ ms}^{-1}$
• B.$\sqrt{31} \text{ ms}^{-1}$
• C.$6\sqrt{3} \text{ ms}^{-1}$
• D.$4\sqrt{7} \text{ ms}^{-1}$
• E.$2\sqrt{43} \text{ ms}^{-1}$
• F.$4\sqrt{13} \text{ ms}^{-1}$
• G.$4\sqrt{19} \text{ ms}^{-1}$
• H.$2\sqrt{127} \text{ ms}^{-1}$

Answer: D

• A.$\frac{2\pi R^3 \rho_0}{3}$
• B.$\frac{5\pi R^3 \rho_0}{6}$
• C.$\frac{8\pi R^3 \rho_0}{9}$
• D.$\pi R^3 \rho_0$
• E.$\frac{29\pi R^3 \rho_0}{24}$
• F.$\frac{19\pi R^3 \rho_0}{15}$
• G.$\frac{4\pi R^3 \rho_0}{3}$
• H.$2\pi R^3 \rho_0$

Answer: B