Question
(a) Underline all the SI base units in the following list.
ampere
coulomb
current
kelvin
newton [1]
\(s=\frac{v^2}{2 a}\)
where a is the acceleration of the car and v is its final velocity.
State two conditions that apply to the motion of the car in order for the above equation to be valid.
1 ………………………………………………………………………………………………………………………………
2 ……………………………………………………………………………………………………………………………… [2]
(c) An experiment is performed to determine the acceleration of the car in (b). The following measurements are obtained:
$
\begin{aligned}
& s=3.89 \mathrm{~m} \pm 0.5 \% \\
& v=2.75 \mathrm{~ms}^{-1} \pm 0.8 \% .
\end{aligned}
$
(i) Calculate the acceleration a of the car.
$
a=
$
$
\mathrm{ms}^{-2}[1]
$
(ii) Determine the percentage uncertainty, to two significant figures, in a.
percentage uncertainty \(=\) \(\%\) [2]
(iii) Use your answers in (c)(i) and (c)(ii) to determine the absolute uncertainty in the calculated value of a.
absolute uncertainty = \(\mathrm{ms}^{-2}\) [1] [Total: 7]
▶️Answer/Explanation
Ans:
Express integrand as \(2 \sec ^2 \frac{1}{2} x-2\)
Integrate to obtain form \(a \tan \frac{1}{2} x+b x\)
Obtain correct \(4 \tan \frac{1}{2} x-2 x\)
Obtain \(4-\pi\)
Question
A motor uses a wire to raise a block, as illustrated in Fig. 2.1.
The base of the block takes a time of 0.49 s to move vertically upwards from level \(X\) to level \(Y\) at a constant speed of \(0.64 \mathrm{~m} \mathrm{~s}^{-1}\). During this time the wire has a strain of 0.0012 . The wire is made of metal of Young modulus \(2.2 \times 10^{11} \mathrm{~Pa}\) and has a uniform cross-section.
The block has a weight of \(1.4 \times 10^4 \mathrm{~N}\). Assume that the weight of the wire is negligible.
(a) Calculate:
(i) the cross-sectional area \(A\) of the wire
$
A=
$
(ii) the increase in the gravitational potential energy of the block for the movement of its base from \(X\) to \(Y\).
(b) The motor has an efficiency of 56%.
Calculate the input power to the motor as the base of the block moves from X to Y.
input power \(=\)
W [3]
(c) The base of the block now has a uniform deceleration of magnitude \(1.3 \mathrm{~m} \mathrm{~s}^{-2}\) from level \(Y\) until the base of the block stops at level \(Z\).
Calculate the tension \(T\) in the wire as the base of the block moves from \(Y\) to \(Z\).
T = ……………………………………………… N [3]
(d) The base of the block is at levels \(\mathrm{X}, \mathrm{Y}\) and \(\mathrm{Z}\) at times \(t_{\mathrm{X}}, t_{\mathrm{Y}}\) and \(t_{\mathrm{Z}}\) respectively.
On Fig. 2.2, sketch a graph to show the variation with time \(t\) of the distance \(d\) of the base of the block from level \(\mathrm{X}\). Numerical values of \(d\) and \(t\) are not required.
▶️Answer/Explanation
Ans:
State \(\frac{\tan \theta-\tan 60}{1+\tan \theta \tan 60}=\frac{3}{\tan \theta}\)
Attempt arrangement of equation to quadratic form
Obtain \(\tan ^2 \theta-4 \sqrt{3} \tan \theta-3=0\)
Solve 3 -term quadratic equation to obtain at least one value of \(\theta\)
Obtain −22.2 and 82.2
Question
A uniform beam AB is attached by a hinge to a wall at end A, as shown in Fig. 3.1.
The beam has length \(0.50 \mathrm{~m}\) and weight \(W\). A block of weight \(12 \mathrm{~N}\) rests on the beam at a distance of \(0.15 \mathrm{~m}\) from end \(\mathrm{B}\).
The beam is held horizontal and in equilibrium by a string attached between end \(\mathrm{B}\) and a fixed point \(\mathrm{C}\). The string has a tension of \(17 \mathrm{~N}\) and is at an angle of \(50^{\circ}\) to the horizontal.
(a) State two conditions for an object to be in equilibrium.
1 ………………………………………………………………………………………………………………………………
2 ………………………………………………………………………………………………………………………………[2]
(b) Show that the vertical component of the tension in the string is 13N. [1]
(c) By taking moments about end A, calculate the weight W of the beam.
W = ……………………………………………… N [2]
(d) Calculate the magnitude of the vertical component of the force exerted on the beam by the
hinge.
force = ……………………………………………… N [1]
(e) The block is now moved closer to end A of the beam. Assume that the beam remains horizontal.
State whether this change will increase, decrease or have no effect on the horizontal
component of the force exerted on the beam by the hinge……………………………………………………………………………………………………………………………. [1][Total: 7]
▶️Answer/Explanation
Ans:
a. Substitute \(x=-2\) and equate to zero
Substitute x =3 and equate to 35
Obtain \(-8 a-4 a-2 a+b=0\) and \(27 a-9 a+3 a+b=35\)
Solve a pair of relevant simultaneous linear equations to find a or b
Obtain a =1 and b =14
b. Divide by \(x+2\) at least as far as the \(x\) term
Obtain \([(x+2)]\left(x^2-3 x+7\right)\)
Conclude with reference to −2 , and discriminant is 9 − 28 and hence no root
Question
Two blocks slide directly towards each other along a frictionless horizontal surface, as shown in Fig. 4.1. The blocks collide and then move as shown in Fig. 4.2.
Block \(X\) initially moves to the right with a momentum of \(0.37 \mathrm{~kg} \mathrm{~ms}^{-1}\). Block \(Y\) initially moves to the left with a momentum of \(0.65 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}\). After the blocks collide, block \(X\) moves to the left back along its original path with a momentum of \(0.13 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}\). Block \(Y\) also moves to the left after the collision.
(a) Block X has an initial kinetic energy of \(0.30 \mathrm{~J}\).
Calculate the mass of block \(X\).
mass = …………………………………………….. kg [3]
(b) Determine the magnitude of the momentum of block Y after the collision.
momentum \(=\) \(\mathrm{kgm} \mathrm{s}^{-1}\) [1]
(c) Block X exerts an average force of 7.7N on block Y during the collision.
Calculate the time that the blocks are in contact with each other.
time = ………………………………………………. s [2] [Total: 6]
▶️Answer/Explanation
Ans:
Draw V-shaped graph with vertex on positive \(x\)-axis
Draw approximately correct graph of \(y=3 x-3\) with greater gradient
b. Attempt solution of linear equation where signs of \(2 x\) and \(3 x\) are different
Solve \(-2 x+11=3 x-3\) to obtain \(x=\frac{14}{5}\)
Conclude \(x>\frac{14}{5}\)
Alternative method for Question 4(b)
Attempt solution of 3-term equation \((2 x-11)^2=(3 x-3)^2\) to obtain at least one value of \(x\)
Obtain at least \(x=\frac{14}{5}\)
Conclude \(x>\frac{14}{5}\)
c.Attempt value of \(N\) (maybe non-integer at this stage) using logarithms and their answer to part (b).
Conclude with single integer 17
Question
On Fig. 5.1, sketch the new trace shown on the screen of the CRO.[3]
The wavelength of the light from the laser is \(630 \mathrm{~nm}\). The light is incident normally on the double slit. The separation of the two slits is \(3.6 \times 10^{-4} \mathrm{~m}\). The perpendicular distance between the double slit and the screen is \(D\).
Coherent light waves from the slits form an interference pattern of bright and dark fringes on the screen. The distance between the centres of two adjacent bright fringes is \(4.0 \times 10^{-3} \mathrm{~m}\). The central bright fringe is formed at point \(P\).
(i) Explain why a bright fringe is produced by the waves meeting at point \(P\).[1]
(ii) Calculate distance \(D\).
D = ………………………………………………m [3]
(c) The wavelength \(\lambda\) of the light in (b) is now varied. This causes a variation in the distance \(x\) between the centres of two adjacent bright fringes on the screen. The distance \(D\) and the separation of the two slits are unchanged.
On Fig. 5.3, sketch a graph to show the variation of \(x\) with \(\lambda\) from \(\lambda=400 \mathrm{~nm}\) to \(\lambda=700 \mathrm{~nm}\). Numerical values of \(x\) are not required.
▶️Answer/Explanation
Ans:
Integrate to obtain form \(k_1 \ln (1+2 x)+k_2 \ln x\)
Obtain correct \(2 \ln (1+2 x)+3 \ln x\)
Use limits correctly and equate to \(\ln 10\)
Apply relevant logarithm properties correctly and arrange as far as \(a^3=\ldots\)
Confirm given result \(a=\sqrt[3]{90(1+2 a)^{-2}}\) with sufficient detail
b.Use iteration process correctly at least once
Obtain final answer 1.68
Show sufficient iterations to \(5 \mathrm{sf}\) to justify answer or show a sign change in the interval \([1.675,1.685]\)
Question
(a) Define the potential difference across a component……………………………………………………………………………………………………………………………. [1]
(b) The variation with potential difference V of the current I in a semiconductor diode is shown inFig. 6.1.
Use Fig. 6.1 to describe qualitatively:
(i) the resistance of the diode in the range \(V=0\) to \(V=0.25 \mathrm{~V}\)[1]
(ii) the variation, if any, in the resistance of the diode as \(V\) changes from \(V=0.75 \mathrm{~V}\) to \(V=1.0 \mathrm{~V}\).[1]
(c) A battery of electromotive force (e.m.f.) 12V and negligible internal resistance is connected to a uniform resistance wire XY, a fixed resistor and a variable resistor, as shown in Fig. 6.2.
The fixed resistor has a resistance of \(5.0 \Omega\). The current in the battery is \(2.7 \mathrm{~A}\) and the current in the fixed resistor is \(1.5 \mathrm{~A}\).
(i) Calculate the current in the resistance wire.
current = ……………………………………………….A [1]
(ii) Determine the resistance of the variable resistor.
resistance \(=\) \(\Omega[2]\)
(iii) Wire XY has a length of 2.0m. Point Z on the wire is a distance of 1.6m from point X.
The fixed resistor is connected to the variable resistor at point W. Determine the potential difference between points W and Z.
potential difference = ……………………………………………… V [3]
(iv) The resistance of the variable resistor is now increased.
By considering the currents in every part of the circuit, state and explain whether the total power produced by the battery decreases, increases or stays the same.[3] [Total: 12]
▶️Answer/Explanation
Ans:
Differentiate using quotient rule
Obtain \(\frac{\left(\mathrm{e}^x+2\right) 8 \mathrm{e}^{2 x}-\left(4 \mathrm{e}^{2 x}+9\right) \mathrm{e}^x}{\left(\mathrm{e}^x+2\right)^2}\)
Substitute \(x=0\) in first derivative and attempt evaluation
Obtain \(\frac{11}{9}\)
b.Equate first derivative to zero and attempt factorisation or equivalent
Solve a three-term quadratic equation in \(\mathrm{e}^x\) to obtain \(\mathrm{e}^x=\ldots\)
Obtain \(x\)-coordinate \(\ln \frac{1}{2}\) or \(-\ln 2\)
Obtain y-coordinate 4
Question
(a) Nuclei \(X\) and \(Y\) are different isotopes of the same element.
Nucleus \(X\) is unstable and emits a \(\beta^{+}\)particle to form nucleus \(Z\).
By comparing the number of protons in each nucleus, state and explain whether the charge of nucleus \(\mathrm{X}\) is less than, the same as or greater than the charge of:
(i) nucleus \(Y\)[1]
(ii) nucleus \(Z\).[2]
(b) Hadrons can be divided into two groups (classes), P and Q. Group P is baryons.
(i) State the name of group \(Q\).[1]
(ii) Describe, in general terms, the quark structure of hadrons that belong to group \(Q\).[1] [Total: 5]
▶️Answer/Explanation
Ans:
Use identity \(\sin 2 t=2 \sin t \cos t\)
Attempt solution of \(y=0\) for \(\cos t\)
Obtain \(\cos t=\frac{2}{3}\)
b.Differentiate to obtain at least one of \(\frac{\mathrm{d} x}{\mathrm{~d} t}\) and \(\frac{\mathrm{d} y}{\mathrm{~d} t}\) correct
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 \cos 2 t-4 \cos t}{k \sec ^2 t}\)
Attempt to express \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) in terms of \(\cos t\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6\left(2 \cos ^2 t-1\right) \cos ^2 t-4 \cos ^3 t}{k}\)
c.Substitute value from part (a) in expression for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) involving \(k\) and \(\cos t\)
Equate to \(-\frac{10}{9}\) and solve for \(k\)
Obtain \(k=\frac{4}{3}\)