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(a) (i) Define power.
.........................................
.........................................[1]
(ii) Show that the SI base units of power are $\text{kg m}^2\text{s}^{-3}$.
.........................................[1]
(b) All bodies radiate energy. The power $P$ radiated by a body is given by
$$P = kAT^4$$
where $T$ is the thermodynamic temperature of the body,
$A$ is the surface area of the body
and $k$ is a constant.
(i) Determine the SI base units of $k$.
base units .........................................[2]
(ii) On Fig. 1.1, sketch the variation with $T^2$ of $P$. The quantity $A$ remains constant.
[Image_1: Fig 1.1]
.........................................[1]
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A liquid of density $\rho$ fills a container to a depth $h$, as shown in Fig. 2.1.
The base of the container has area $A$.
(a) Derive, from the definitions of pressure and density, the equation
$p = \rho gh$
where $p$ is the pressure exerted by the liquid on the base of the container and $g$ is the acceleration of free fall.
(b) A small solid sphere falls with constant velocity through the liquid.
(i) State
1. the names of the three forces acting on the sphere,
2. a word equation that relates the magnitudes of these forces. [2]
(ii) State and explain the changes in energy that occur as the sphere falls. [2]
(c) The liquid in the container is liquid L. Liquid M is now added to the container. The two liquids do not mix. The total depth of the liquids is 0.17 m.
Fig. 2.2 shows how the pressure $p$ inside the liquids varies with height $x$ above the base of the container.
Use Fig. 2.2 to
(i) state the value of atmospheric pressure, [1]
(ii) determine the density of liquid M. [2]
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(a) State the principle of conservation of momentum. [2]
(b) Ball A moves with speed $v$ along a horizontal frictionless surface towards a stationary ball B, as shown in Fig. 3.1.
Ball A has mass 4.0 kg and ball B has mass 12 kg.
The balls collide and then move apart as shown in Fig. 3.2.
Ball A has velocity 6.0 m s$^{-1}$ at an angle of $\theta$ to the direction of its initial path.
Ball B has velocity 3.5 m s$^{-1}$ at an angle of 30° to the direction of the initial path of ball A.
(i) By considering the components of momentum at right-angles to the direction of the initial path of ball A, calculate $\theta$. [3]
(ii) Use your answer in (i) to show that the initial speed $v$ of ball A is 12 m s$^{-1}$.
Explain your working. [2]
(iii) By calculation of kinetic energies, state and explain whether the collision is elastic or inelastic. [3]
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(a) By reference to the direction of propagation of energy, explain what is meant by a longitudinal wave. [1]
(b) A car horn emits a sound wave of frequency 800 Hz. A microphone and a cathode-ray oscilloscope (c.r.o.) are used to analyse the sound wave. The waveform displayed on the c.r.o. screen is shown in Fig. 4.1.
Determine the time-base setting, in s cm^{-1}, of the c.r.o. [3]
(c) The intensity I of the sound at a distance r from the car horn in (b) is given by the expression
$$I = \frac{k}{r^2}$$
where k is a constant.
Fig. 4.2 shows the car in (b) on a road.
An observer stands at point O. Initially the car is parked at point X which is 120 m away from point O. The car then moves directly towards the observer and stops at point Y, a distance of 30 m away from O.
The car horn continuously emits sound when the car is moving between points X and Y.
(i) The sound wave at point O has amplitude A_X when the car is at X and has amplitude A_Y when the car is at Y.
Calculate the ratio \(\frac{A_Y}{A_X}\). [3]
(ii) When the car is parked at X, the frequency of the sound from the horn that is detected by the observer is 800 Hz. As the car moves from X to Y, the maximum change in the detected frequency is 16 Hz. The speed of the sound in air is $330 m s^{-1}$.
Determine, to two significant figures,
- the minimum wavelength of the sound detected by the observer, [2]
- the maximum speed of the car. [2]
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(a) Define electric field strength.
(b) Two parallel metal plates in a vacuum are separated by 0.045 m. A potential difference $V$ is applied between the plates, as shown in Fig. 5.1.
A proton is initially at rest on the surface of the positive plate. The proton in the uniform electric field takes a time of $1.5 \times 10^{-7}$ s to reach the negative plate.
(i) Show that the acceleration of the proton is $4.0 \times 10^{12}$ m s$^{-2}$. [2]
(ii) Calculate the electric force on the proton.
force = ..................................................... N [1]
(iii) Use your answer in (ii) to determine
1. the electric field strength,
field strength = ......................................... N C$^{-1}$ [2]
2. the potential difference $V$ between the plates.
$V$ = ....................................................... V [2]
(c) An $\alpha$ particle is now accelerated between the two metal plates in (b) by the electric field.
Calculate the ratio $$\frac{\text{acceleration of } \alpha \text{ particle}}{\text{acceleration of proton}}.$$
ratio = ...................................................... [2]
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A filament lamp is rated as 30 W, 120 V. A potential difference of 120 V is applied across the lamp.
(a) For the filament wire of the lamp, calculate
(i) the current,
current = ............................................................... A [2]
(ii) the number of electrons passing a point in 3.0 hours.
number = ............................................................... [2]
(b) Show that the resistance of the filament wire is 480Ω.
[2]
(c) The filament wire has an uncoiled length of 580 mm and is made of metal. The metal has resistivity $6.1 \times 10^{-7} \Omega \text{m}$ at the operating temperature of the lamp.
Calculate the diameter of the wire.
diameter = ............................................................... m [3]
(d) The potential difference across the lamp is now reduced. State and explain the effect, if any, on the resistance of the filament wire.
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(a) A nucleus X decays by emitting a $\beta^+$ particle to form a new nucleus, $^{23}_{11}\text{Na}$.
State the number of nucleons and the number of neutrons in nucleus X.
number of nucleons = ...........................................................
number of neutrons = ...........................................................
[2]
(b) State one similarity and one difference between a $\beta^+$ particle and a $\beta^-$ particle.
similarity: ......................................................................
difference: ......................................................................
[2]
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