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(a) State what is meant by work done. [1]
(b) Use the answer to (a) to determine the SI base units of power. [2]
(c) The maximum useful output power $P$ of a car travelling on a horizontal road is given by
$$P = v^3 b$$
where $v$ is the maximum speed of the car and $b$ is a constant.
For the car,
$P = 84 \text{kW} \pm 5\%
and $b = 0.56 \pm 7\%$ in SI units.
(i) Calculate the value of $v$. [2]
(ii) Determine the absolute uncertainty in the value of $v$. [2]
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A spherical balloon is filled with a fixed mass of gas. A small block is connected by a string to the balloon, as shown in figure 1.
The block is held on the ground by an external force so that the string is vertical. The density of the air surrounding the balloon is $1.2\, \text{kg/m}^3$. The upthrust acting on the balloon is $0.071\, \text{N}$. The upthrust acting on the string and block is negligible.
(a) By using Archimedes' principle, calculate the radius $r$ of the balloon. [2]
(b) The total weight of the balloon, string and block is $0.053\, \text{N}$.
The external force holding the block on the ground is removed so that the released block is lifted vertically upwards by the balloon.
Calculate the acceleration of the block immediately after it is released. [3]
(c) The balloon continues to lift the block. The string breaks as the block is moving vertically upwards with a speed of $1.4\, \text{m/s}$. After the string breaks, the detached block briefly continues moving upwards before falling vertically downwards to the ground. The block hits the ground with a speed of $3.6\, \text{m/s}$.
Assume that the air resistance on the block is negligible.
(i) By considering the motion of the block after the string breaks, calculate the height of the block above the ground when the string breaks. [2]
(ii) The string breaks at time $t = 0$ and the block hits the ground at time $t = T$.
On Figure 2, sketch a graph to show the variation of the velocity $v$ of the block with time $t$ from $t = 0$ to $t = T$.
Numerical values of $t$ are not required. Assume that $v$ is positive in the upward direction. [2]
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(a) State what is meant by the centre of gravity of an object. [1]
(b) A uniform beam AB is attached by a frictionless hinge to a vertical wall at end A. The beam is held so that it is horizontal by a metal wire CD, as shown in Fig. 3.1.
The beam is of length 0.96 m and weight 23 N. A block of weight $W$ rests on the beam at a distance of 0.20 m from end B. The wire is attached to the beam at point D which is a distance of 0.40 m from end B. The wire exerts a force on the beam of 45 N at an angle of 37° to the horizontal. The beam is in equilibrium.
(i) Calculate the vertical component of the force exerted by the wire on the beam. [1]
(ii) By taking moments about A, calculate the weight $W$ of the block. [3]
(iii) The hinge exerts a force on the beam at end A.
Calculate the horizontal component of this force. [1]
(iv) The block is now placed closer to point D on the beam.
State whether this change will increase, decrease or have no effect on the tension in the wire. [1]
(v) The stress in the wire is $5.3 \times 10^7$ Pa. The wire is now replaced by a second wire that has a radius which is three times greater than that of the original wire. The tension in the wire is unchanged.
Calculate the stress in the second wire. [2]
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A horizontal spring is fixed at one end. A block is pushed against the other end of the spring so that the spring is compressed, as shown in Fig. 4.1.
The block is released and accelerates along a horizontal frictionless surface as the spring returns to its original length. The block leaves the end of the spring with a speed of 2.3 m s-1, as shown in Fig. 4.2.
The block has a mass of 250 g and the spring has a spring constant of 420 N m-1.
Assume that the spring always obeys Hooke's law and that all the elastic potential energy of the spring is transferred to the kinetic energy of the block.
(a) Calculate the kinetic energy of the block as it leaves the spring. [2]
(b) Calculate the compression of the spring immediately before the block is released. [2]
(c) After leaving the spring, the block moves along the surface until it hits a barrier at a speed of 2.3 m s-1. The block then rebounds at a speed of 1.5 m s-1 and moves back along its original path. The block is in contact with the barrier for a time of 0.086 s.
Calculate:
(i) the change in momentum of the block during the collision [2]
(ii) the average resultant force exerted on the block during the collision. [1.
(d) The maximum compression x of the spring is now varied in order to vary the kinetic energy $E_K$ of the block as it leaves the spring. Assume that all the elastic potential energy in the spring is always transferred to the kinetic energy of the block.
On Fig. 4.3, sketch a graph to show the variation with x of $E_K$. [1]
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(a) Two progressive sound waves meet to form a stationary wave. The two waves have the same amplitude, wavelength, frequency and speed.
State the other condition that must be fulfilled by the two waves in order for them to produce the stationary wave. [1]
(b) A stationary wave is formed on a string that is stretched between two fixed points A and B. Fig. 5.1 shows the string at time $t = 0$ when each point is at its maximum displacement.
Distance AB is 0.80 m. The period of the stationary wave is 0.016 s.
(i) On Fig. 5.1, sketch a solid line to show the position of the string: [2]
- at time $t = 0.004$ s (label this line P)
- at time $t = 0.024$ s (label this line Q).
(ii) Determine the speed of a progressive wave along the string. [3]
(c) A beam of vertically polarised light of intensity $I_0$ is incident normally on a polarising filter that has its transmission axis at 30° to the vertical, as shown in Fig. 5.2.
The transmitted light from the first polarising filter has intensity $I_1$. This light is then incident normally on a second polarising filter that has its transmission axis at 90° to the vertical. The transmitted light from the second filter has intensity $I_2$.
Calculate:
(i) the ratio $\frac{I_1}{I_0}$ [2]
(ii) the ratio $\frac{I_2}{I_0}$ [2]
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(a) Define electric potential difference.
.................................................................................................................................
................................................................................................................................. [1]
(b) A battery is connected to two resistors X and Y, as shown in Fig. 6.1.
The resistance of resistor X is greater than the resistance of resistor Y.
State and explain which resistor dissipates more power.
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
................................................................................................................................. [3]
(c) A battery of electromotive force (e.m.f.) 9.0V and internal resistance $r$ is connected to two resistors P and Q, as shown in Fig. 6.2.
A total charge of 650C moves through resistor P in a time interval of 540s. During this time resistor P dissipates 4800J of energy. The current in resistor Q is 4.5A. Assume that the e.m.f. of the battery remains constant.
Calculate:
(i) the current in resistor P
current = ........................................................... A [2]
(ii) the potential difference across resistor P
potential difference = ........................................................... V [2]
(iii) the internal resistance $r$ of the battery.
$r$ = ........................................................... \Omega [2]
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(a) Describe the structure of an \textbf{atom} of uranium-238, $^{238}_{92}\text{U}$.
..............................................................................................................................
..............................................................................................................................
.............................................................................................................................. [2]
(b) The decay of uranium-238 is shown by the equation
$$^{238}_{92}\text{U} \rightarrow ^{234}_{90}\text{Th} + \text{X}.$$
For nucleus X, calculate the ratio, in C kg$^{-1}$, of
$$\frac{\text{charge}}{\text{mass}}.$$
ratio = .............................................~C kg$^{-1}$ [3]
(c) Two particles P and Q each consist of three quarks. These quarks are up (u) or down (d) quarks.
Particle P has no overall charge.
Particle Q has an overall charge of +2e, where e is the elementary charge.
State the quark composition of:
(i) particle P
............................................................................................................................ [1]
(ii) particle Q.
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