(a) In his first attempt, a long jumper took off from the springboard with a speed of 8 ms\(^{-1}\) at 30° to the horizontal. He makes a second attempt with the same speed at 45° to the horizontal. Given that the expression for the horizontal range of a projectile is \(\frac{v^2 sin \theta}{g}\) where all the symbols have their usual meanings, show that he gains a distance of 0.8576 m in his second attempt.
(b)(i) State Hooke's law of elasticity.
(ii) Describe an experiment to verify Hooke's law.
(iii) State two precautions you would take if you were to perform this experiment in the laboratory.
(c) A spiral spring of natural length 20.00 cm has a scale pan hanging freely in its lower end. When an object of mass 40 g is placed in the pan, its length becomes 21.80 cm. When another object of mass 60g Is placed in the pan, the length becomes 22.05cm. Calculate the mass of the scale pan. [g = 10 ms\(^{-2}\)]
(a) Gain in the long jump
The horizontal range of a projectile is \(R = \dfrac{v^2\sin 2\theta}{g}\). With \(v = 8\,\text{ms}^{-1}\) and \(g = 10\,\text{ms}^{-2}\), \(\dfrac{v^2}{g} = \dfrac{64}{10} = 6.4\,\text{m}\).
First attempt (\(\theta = 30^\circ\)): \(R_1 = 6.4\sin 60^\circ = 6.4(0.8660) = 5.5424\,\text{m}\).
Second attempt (\(\theta = 45^\circ\)): \(R_2 = 6.4\sin 90^\circ = 6.4(1) = 6.4000\,\text{m}\).
\[ R_2 - R_1 = 6.4000 - 5.5424 = 0.8576\,\text{m} \quad\text{(shown)} \]
(b)(i) Hooke's law: Provided the elastic limit is not exceeded, the extension of an elastic material is directly proportional to the applied force (load).
(b)(ii) Experiment to verify Hooke's law: A spiral spring is clamped vertically with a pointer against a metre rule. The unstretched pointer reading is noted. Known masses are added one at a time; for each load the new pointer reading is taken and the extension found by subtraction. A graph of load against extension is plotted; a straight line through the origin verifies that extension is proportional to load, confirming Hooke's law.
(b)(iii) Two precautions:
- Do not exceed the elastic limit of the spring (avoid overloading so it returns to its original length).
- Take readings with the eye level to the pointer to avoid parallax error; allow the spring to settle before reading.
(c) Mass of the scale pan
Extension is proportional to the total load (pan mass \(m\) plus object). Natural length \(= 20.00\,\text{cm}\).
With 40 g: length \(21.80\,\text{cm}\), extension \(e_1 = 1.80\,\text{cm}\), load \(=(m+40)\).
With 60 g: length \(22.05\,\text{cm}\), extension \(e_2 = 2.05\,\text{cm}\), load \(=(m+60)\).
\[ \frac{m+40}{1.80} = \frac{m+60}{2.05} \]
\[ 2.05(m+40) = 1.80(m+60) \]
\[ 2.05m + 82 = 1.80m + 108 \;\Rightarrow\; 0.25m = 26 \;\Rightarrow\; m = 104\,\text{g} \]
The mass of the scale pan is \(104\,\text{g}\ (0.104\,\text{kg})\).
