WAEC SSCE - Physics - 2007 (Practical)

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Practical: deviation through a prism with a plane mirror on face AC

The prism is placed on its traced outline ABC and light is aimed at the point O on face AB at an angle of incidence \(i\). The plane mirror is held against face AC. For each value of \(i\) the emergent direction is fixed by the no-parallax pin method: pins \(R_1\) and \(R_2\) are set on the incident line NO, and pins \(R_3\) and \(R_4\) are set while viewing through face BC. Line \(R_4R_3\) is produced to meet NO produced at T, where the angle \(\theta\) is measured, and the angle \(e\) is measured at D. The construction is shown below.

Table of readings (observations)

S/N	i / °	θ / °	e / °
1	25	62	28
2	30	54	32
3	35	45	35
4	40	40	37
5	45	30	41

Graph of e against \(\theta\)

A graph of \(e\) (vertical axis) against \(\theta\) (horizontal axis) is plotted, and a straight line of best fit is drawn through the points.

Slope and evaluation of k

Two widely separated points are read from the line of best fit: \((\theta_1,e_1)=(10,49)\) and \((\theta_2,e_2)=(83,20)\).

\[ s=\frac{\Delta e}{\Delta\theta}=\frac{20-49}{83-10}=\frac{-29}{73}=-0.397 \]

The magnitude of the slope is \(|s|=0.397\) and the intercept on the \(e\)-axis is \(c=53\). Therefore

\[ k=s^{-1}=\frac{1}{0.397}=2.52 \]

Two precautions

• The plane mirror was pressed firmly against face AC so that it did not shift while the emergent pins were being fixed.
• All pins were fixed vertically and spaced about 4 cm apart, and viewing was done with the eye low so that parallax between the pins and their images was eliminated.

(b)(i) Four characteristics of the image formed by a plane mirror

• It is virtual (cannot be received on a screen).
• It is erect (upright).
• It is the same size as the object.
• It is laterally inverted and is the same distance behind the mirror as the object is in front.

(b)(ii) Conditions for total internal reflection

• Light must be travelling from an optically denser medium to an optically less dense medium.
• The angle of incidence in the denser medium must be greater than the critical angle for the two media.

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Measurement of e.m.f. of the accumulator

E.m.f. of the accumulator, \(E = 2.0\ \text{V}\).

Observation and table of readings

For each setting the resistance in \(RB_1\) was made equal to the resistance in \(RB_2\) (\(R\) in \(RB_1 = R\) in \(RB_2 = R\)). With \(K_1\) and \(K_2\) closed the potential difference \(V_1\) across the standard resistor \(R_x\) was read, and \(V_1^{-1}\) evaluated.

\(R\ (\Omega)\)	\(V_1\ (\text{V})\)	\(V_1^{-1}\ (\text{V}^{-1})\)
1	1.50	0.667
2	1.30	0.769
3	1.15	0.870
4	1.10	0.909
5	0.90	1.111

Graph of \(V_1^{-1}\) against \(R\)

The graph of \(V_1^{-1}\) (vertical axis) against \(R\) (horizontal axis), with both axes starting from the origin, is a rising straight line.

Slope and intercept

Taking two widely spaced points on the line of best fit, \((0.3,\ 0.60)\) and \((6.5,\ 1.22)\):

\[ s = \frac{\Delta (V_1^{-1})}{\Delta R} = \frac{1.22 - 0.60}{6.5 - 0.3} = \frac{0.62}{6.2} = 0.10\ \text{V}^{-1}\,\Omega^{-1} \]

The intercept on the vertical axis (value of \(V_1^{-1}\) where the line cuts \(R = 0\)) is

\[ c = 0.57\ \text{V}^{-1}. \]

Evaluation of \(y = \dfrac{1}{s}\)

\[ y = \frac{1}{s} = \frac{1}{0.10} = 10\ \text{V}\,\Omega. \]

Two precautions

1. I opened the keys when readings were not being taken, to avoid heating of the resistors and drift in the cell.
2. I made clean, tight connections at all terminals and read the voltmeter with my eye directly in front of the pointer to avoid parallax error (also correcting for any zero error).

(b)(i) Potential difference between two points

The potential difference between two points in an electric circuit is the work done (in joules) in moving one coulomb of positive charge from one point to the other. It is measured in volts.

(b)(ii) Internal resistance of the cell

Given \(E = 3\ \text{V}\), \(R = 4\ \Omega\) and current \(I = 0.5\ \text{A}\):

\[ E = IR + Ir \] \[ 3 = (0.5 \times 4) + 0.5r \] \[ 3 = 2 + 0.5r \] \[ 0.5r = 1 \] \[ r = \frac{1}{0.5} = 2\ \Omega. \]

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Simple pendulum: graph of \(T^{2}\) against \(L\)

The pendulum obeys \(T = 2\pi\sqrt{\dfrac{l}{g}}\), so that \(T^{2} = \dfrac{4\pi^{2}}{g}\,l\). Here the length used for the horizontal axis is \(L = l - 30\ \text{cm}\), and the time \(t\) is measured for 20 complete oscillations, giving \(T = \dfrac{t}{20}\).

Table of readings

S/N	l (cm)	t (s) for 20 osc.	T (s)	T² (s²)	L = l − 30 (cm)
1	130	45.70	2.29	5.22	100
2	110	42.00	2.10	4.41	80
3	90	38.00	1.90	3.61	60
4	70	33.10	1.66	2.74	40
5	50	28.70	1.44	2.06	20

Graph of \(T^{2}\) against \(L\)

Slope and intercept

Reading two widely separated points on the line of best fit, \((L_{1}, T_{1}^{2}) = (100,\ 5.22)\) and \((L_{2}, T_{2}^{2}) = (20,\ 2.06)\):

\[ s = \frac{\Delta T^{2}}{\Delta L} = \frac{5.22 - 2.06}{100 - 20} = \frac{3.16}{80} = 0.039\ \text{s}^{2}\,\text{cm}^{-1} \]

The line cuts the \(T^{2}\) axis at

\[ C = 1.3\ \text{s}^{2}. \]

Evaluation

Taking \(\pi = \dfrac{22}{7}\):

i. \[ k_{1} = \frac{4\pi^{2}}{s} = \frac{4\left(\frac{22}{7}\right)^{2}}{0.039} = \frac{4 \times 9.878}{0.039} = \frac{39.51}{0.039} = 1013\ \text{cm s}^{-2} = 10.13\ \text{m s}^{-2}. \] This is the acceleration of free fall due to gravity, \(g\).

ii. \[ k_{2} = \frac{C}{s} = \frac{1.3}{0.039} = 33.33. \]

Two precautions

1. I displaced the bob through a small angle and released it so that it swung in one vertical plane only, avoiding conical (elliptical) oscillations.
2. I avoided parallax error by reading the metre rule and stop-clock with the line of sight perpendicular to the scale, and counted the oscillations from the central rest position.

(b)

i. The period of oscillation of a body is the time taken for the body to make one complete to-and-fro movement (one full oscillation).

ii. When a body falls freely under its weight alone, the rate of increase of its velocity with time, caused by the Earth's gravitational pull, is the acceleration of free fall due to gravity. It is directed vertically downwards and has a value of about \(10\ \text{m s}^{-2}\) near the Earth's surface.