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(a) Determine the SI base units of the moment of a force. [1]
(b) A uniform square sheet of card ABCD is freely pivoted by a pin at a point P. The card is held in a vertical plane by an external force in the position shown in figure.
The card has weight 0.15 N which may be considered to act at the centre of gravity G. Each side of the card has length 17 cm. Point P lies on the horizontal line AC and is 4.0 cm from corner A. Line BD is vertical.
The card is released by removing the external force. The card then swings in a vertical plane until it comes to rest.
(i) Calculate the magnitude of the resultant moment about point P acting on the card immediately after it is released. [2]
(ii) Explain why, when the card has come to rest, its centre of gravity is vertically below point P. [2]
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(a) State what is meant by work done. [1]
(b) A lift (elevator) of weight 13.0kN is connected by a cable to a motor, as shown in Fig. 2.1.
The lift is pulled up a vertical shaft by the cable. A constant frictional force of 2.0kN acts on the lift when it is moving. The variation with time $t$ of the speed $v$ of the lift is shown in Fig. 2.2.
(i) Use Fig. 2.2 to determine:
1. the acceleration of the lift between time $t = 0$ and $t = 3.0\,s$. [2]
2. the work done by the motor to raise the lift between time $t = 3.0\,s$ and $t = 6.0\,s$. [2]
(ii) The motor has an efficiency of 67%. The tension in the cable is $1.6 \times 10^4\,\text{N}$ at time $t = 2.5\,s$.
Determine the input power to the motor at this time. [3]
(iii) State and explain whether the increase in gravitational potential energy of the lift from $t = 0$ to $t = 7.0\,s$ is less than, the same as, or greater than the work done by the motor. A calculation is not required. [1]
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(a) State the property of an object that experiences a force when the object is placed in:
(i) a gravitational field [1]
(ii) an electric field. [1]
(b) A potential difference of $1.2 \times 10^3\, \text{V}$ is applied between a pair of horizontal metal plates in a vacuum, as shown in Fig. 3.1.
The separation of the plates is 3.6 cm. The electric field between the plates is uniform.
A particle of mass $5.9 \times 10^{-6}\, \text{kg}$ and charge $-4.2 \times 10^{-9}\, \text{C}$ enters the field at point X with a horizontal velocity of $0.75 \text{ms}^{-1}$ along a line midway between the two plates.
The particle is deflected by the field and hits the top plate at point Y.
(i) Calculate the magnitude of the electric force acting on the particle in the field. [3]
(ii) By considering the resultant vertical force acting on the particle, show that the acceleration of the particle in the electric and gravitational fields is $14 \text{ms}^{-2}$. [4]
(iii) Determine:
1. the time taken for the particle to move from X to Y [2]
2. the distance $p$ of point Y from the left-hand edge of the top plate. [1]
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A ball X moves along a horizontal frictionless surface and collides with another ball Y, as illustrated in Fig. 4.1.
Ball X has mass 0.300 kg and initial velocity $v_x$ at an angle of 60.0° to line AB.
Ball Y has mass 0.200 kg and initial velocity $6.00 \, \text{ms}^{-1}$ at an angle of 60.0° to line AB.
The balls stick together during the collision and then travel along line AB, as illustrated in Fig. 4.2.
(a) (i) Calculate, to three significant figures, the component of the initial momentum of ball Y that is perpendicular to line AB. [2]
(ii) By considering the component of the initial momentum of each ball perpendicular to line AB, calculate, to three significant figures, $v_x$. [1]
(iii) Show that the speed of the two balls after the collision is $2.4 \, \text{ms}^{-1}$. [2]
(b) The two balls continue moving together along the horizontal frictionless surface towards a spring, as illustrated in Fig. 4.3.
The balls hit the spring and remain stuck together as they decelerate to rest. All the kinetic energy of the balls is converted into elastic potential energy of the spring.
The energy $E$ stored in the spring is given by $$ E = \frac{1}{2}kx^2 $$ where $k$ is the spring constant of the spring and $x$ is its compression.
The spring obeys Hooke’s law and has a spring constant of $72 \, \text{N m}^{-1}$.
(i) Determine the maximum compression of the spring caused by the two balls. [3]
(ii) On Fig. 4.4, sketch graphs to show the variation with compression $x$ of the spring, from zero to maximum compression, of:
- the magnitude of the deceleration $a$ of the balls
- the kinetic energy $E_k$ of the balls.
Numerical values are not required. [3]
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(a) Light waves emerging from the slits of a diffraction grating are coherent and produce an interference pattern.
Explain what is meant by:
(i) coherence [1]
(ii) interference. [1]
(b) A narrow beam of light from a laser is incident normally on a diffraction grating, as shown in Fig. 5.1.
Spots of light are seen on a screen positioned parallel to the grating. The angle corresponding to each of the $\textit{second}$ order maxima is 51°. The number of lines per unit length on the diffraction grating is $6.7 \times 10^5 \text{m}^{-1}$.
(i) Determine the wavelength of the light. [2]
(ii) State and explain the change, if any, to the distance between the second order maximum spots on the screen when the light from the laser is replaced by light of a shorter wavelength. [1]
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A battery of electromotive force (e.m.f.) 12 V and negligible internal resistance is connected to a network of two lamps and two resistors, as shown in Fig. 6.1.
[Image_1: Diagram of the circuit]
The two lamps in the circuit have equal resistances. The two resistors have resistances $R$ and 28 Ω. The lamps are connected at junction X and the resistors are connected at junction Y. The current in the battery is 0.50 A and the current in the lamps is 0.20 A.
(a) Calculate:
(i) the resistance of each lamp
resistance = ......................................................... Ω [2]
(ii) resistance $R$.
$R$ = ......................................................... Ω [2]
(b) Determine the potential difference $V_{XY}$ between points X and Y.
$V_{XY}$ = ......................................................... V [3]
(c) Calculate the ratio
$\frac{\text{total power dissipated by the lamps}}{\text{total power produced by the battery}}$
ratio = ......................................................... [2]
(d) The resistor of resistance $R$ is now replaced by another resistor of lower resistance.
State and explain the effect, if any, of this change on the ratio in (c).
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................ [2]
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A stationary nucleus of a radioactive isotope X decays by emitting an \( \alpha \)-particle to produce a nucleus of neptunium-237 and 5.5 MeV of energy. The decay is represented by
\[ X \rightarrow \ ^{237}_{93}\text{Np} + \alpha + 5.5 \text{MeV}. \]
(a) Calculate the number of protons and the number of neutrons in a nucleus of X.
number of protons = .................................................................
number of neutrons = ............................................................... [2]
(b) Explain why the energy transferred to the \( \alpha \)-particle as kinetic energy is less than the 5.5 MeV of energy released in the decay process.
..........................................................................................................................................................................
.......................................................................................................................................................................... [1]
(c) A sample of X is used to produce a beam of \( \alpha \)-particles in a vacuum. The number of \( \alpha \)-particles passing a fixed point in the beam in a time of 30 s is \( 6.9 \times 10^{11} \).
(i) Calculate the average current produced by the beam of \( \alpha \)-particles.
current = .......................................................... A [2]
(ii) Determine the total power, in W, that is produced by the decay of \( 6.9 \times 10^{11} \) nuclei of X in a time of 30 s.
power = ....................................................... W [2]
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