Questions
PHYSICS PAPER III (ALTERNATIVE A)
Time Allowed: 3 hours
Instructions: Answer Question 1 (compulsory) and ONE other assigned question. Record all observations as made. Only scientific calculator allowed.
1A. A simple example of exponential decay in physics is radioactive decay. A decay of a particular radioactive nucleus cannot be predicted, but for a given radioisotope, we can say that an individual nucleus has a probability λ (called the decay constant) of decaying in unit time (t). Assume the activity (R) of a mixture of radioactive isotope is related to vary according to the law:
R = R₀e^(−λt) ...(1)
where R₀ and λ are constants.
Table 1A presents some experimental results carried out at the Centre for Energy Research and Training.
| t/s | 0.17 | 0.21 | 0.23 | 0.26 | 0.29 | 0.33 | 0.38 | 0.44 | 0.56 | 0.63 |
|---|---|---|---|---|---|---|---|---|---|---|
| R/s⁻¹ | 12.33 | 10.64 | 9.59 | 8.20 | 7.93 | 6.93 | 5.75 | 5.40 | 4.13 | 3.91 |
i. Transform equation (1) into a straight-line graph to determine R₀ and λ.
ii. Form a suitable composite table containing all the necessary parameters to determine the approximate values of R₀ and λ graphically.
iii. Use your graph to determine the approximate values of R₀ and λ.
iv. From the graph, find the approximate value of R when t = 0.36 seconds.
v. From your values of R₀ and λ, use equation (1) to find the value of t when R = 8.99 s⁻¹.
b. If x = 1.3 ± 0.002 and y = 2.17 ± 0.0009, find the error in determining the value of the term x² − √y.
2A. Hang two spiral springs of nearly the same elastic constants in series from a rigid support as shown in Figure 1 with M = 50 g. Displace the spring vertically downward through a small distance and gently release it. The spring performs simple harmonic oscillations. With the help of a stopwatch record the time for 20 complete oscillations. Find the time period T. Evaluate T². Repeat the experiment for five more different values of M within the elastic limit.
i. Tabulate your readings.
ii. Plot a graph of T² against M. Find the slope S of the graph.
iii. If T = 2π√(M/2k), use your slope to find the value of k.
iv. State two precautions taken to ensure accurate results.
3A. Use the diagram in Figure 2.1 to carry out the following instructions. For the ray PQRS, the angle of incidence is i and the angle of emergence is θ.
i. Trace the outline of a triangular glass prism ABC. Set the angle of incidence i = 25°. With the help of the optical pins, find the angle of emergence θ.
ii. Measure and record the corresponding angle marked 'a'. Evaluate sin θ and sin a.
iii. Repeat the experiments with i = 30°, 40°, 45°, 50° and 55° respectively.
iv. Prepare a composite table for your readings. Plot a graph of sin a against sin θ. Find the slope S of the graph and evaluate S⁻¹.
v. State two precautions taken to ensure accurate results. Attach the traces to your answer booklet.
b. Assume the angle of a triangular glass prism of index of refraction 1.73 is approximately 60°. Find the angle of minimum deviation.
Solutions
1A i. Taking natural log of both sides of R = R₀e^(−λt):
ln R = ln R₀ − λt
Plotting ln R (y-axis) against t (x-axis) gives a straight line with slope = −λ and y-intercept = ln R₀.
1A ii. Composite table:
| t/s | 0.17 | 0.21 | 0.23 | 0.26 | 0.29 | 0.33 | 0.38 | 0.44 | 0.56 | 0.63 |
|---|---|---|---|---|---|---|---|---|---|---|
| R/s⁻¹ | 12.33 | 10.64 | 9.59 | 8.20 | 7.93 | 6.93 | 5.75 | 5.40 | 4.13 | 3.91 |
| ln R | 2.51 | 2.36 | 2.26 | 2.10 | 2.07 | 1.94 | 1.75 | 1.69 | 1.42 | 1.36 |
1A iii. Slope using end points (0.17, 2.51) and (0.63, 1.36):
slope = (1.36 − 2.51)/(0.63 − 0.17) = −1.15/0.46 ≈ −2.50 s⁻¹
So λ ≈ 2.50 s⁻¹
Intercept: ln R₀ = 2.51 + 2.50(0.17) = 2.93 → R₀ = e^2.93 ≈ 18.7 s⁻¹
1A iv. At t = 0.36: ln R = 2.93 − 2.50(0.36) = 2.03 → R = e^2.03 ≈ 7.6 s⁻¹
1A v. R = 8.99: ln(8.99) = 2.20
2.20 = 2.93 − 2.50t
t = 0.73/2.50 ≈ 0.29 s
b. x² = 1.69, d(x²) = 2x·dx = 2(1.3)(0.002) = 0.0052
√y = 1.4731, d(√y) = dy/(2√y) = 0.0009/2.9462 = 0.00031
z = x² − √y = 0.2169
dz = d(x²) + d(√y) = 0.0052 + 0.0003 = 0.0055
z = 0.217 ± 0.006
2A i–ii. (Requires actual lab readings — time 20 oscillations at 6 values of M, compute T = time/20, then T², plot T² vs M.)
2A iii. T² = 4π²M/(2k) → slope S = 4π²/(2k) = 2π²/k
k = 2π²/S
2A iv.
- Avoid parallax error by ensuring the eye is level with the reference/scale mark when reading extensions or the pointer position.
- Ensure the spring oscillates only in the vertical plane, avoiding sideways swinging that would introduce timing errors.
3A i–iv. (Requires actual ray-tracing readings for each value of i; tabulate i, θ, a, sin θ, sin a; plot sin a vs sin θ; slope S and S⁻¹ read from graph.)
3A v.
- Use sharp pencils and well-separated optical pins to minimize parallax error in sighting the ray.
- Hold the prism firmly in place so its outline does not shift while tracing the rays.
3A b. n = sin[(A+D)/2] / sin(A/2)
1.73 = sin[(60+D)/2] / sin 30°
sin[(60+D)/2] = 1.73 × 0.5 = 0.865
(60+D)/2 = sin⁻¹(0.865) ≈ 59.9°
60 + D = 119.8°
D ≈ 59.8°