Q#2 (Past Exam Paper – June 2014 Paper 41 & 43 Q4)
A student investigates the energy changes of a mass oscillating on a vertical spring, as shown in Fig. 4.1.
Fig. 4.1
The student draws a graph of the variation with displacement x of energy E of the oscillation, as shown in Fig. 4.2.
Fig. 4.2
(a) State whether the energy E represents the total energy, the potential energy or the kinetic energy of the oscillations. [1]
(b) The student repeats the investigation but with a smaller amplitude. The maximum value of E is now found to be 1.8 mJ.
Use Fig. 4.2 to determine the change in the amplitude. Explain your working. [3]
Solution:
(a) Kinetic energy / KE / EK
{At the maximum displacement, the energy is entirely gravitational potential. The mas is momentarily at rest and thus, its kinetic energy is zero.}
(b)
{Initially, the maximum energy was 2.4 mJ and now, it is 1.8 mJ.
So, change in maximum energy = 2.4 – 1.8 = 0.6 mJ}
EITHER Change in energy = 0.60 mJ
OR The maximum value of E is proportional to (amplitude)2 / equivalent numerical working
{The maximum value of (kinetic energy) E from the graph is the same as the total energy.
Total energy Etotal = ½ mω2x02
The total energy is proportional to the square of the amplitude.
Method 1 (based on calculation):
From the graph,
2.4 mJ corresponds to an amplitude of 1.5 cm.
Since Etotal ∝ x02,
2.4 ∝ 1.52 ------------------ (1)
For 1.8 mJ, let the new amplitude be a.
1.8 ∝ a2 ----------------- (2)
Take (2) divide by (1),
2.4 / 1.8 = 1.52 / a2
giving
a2 = (1.8/2.4) × 1.52
a = 1.3 cm}
The new amplitude is 1.3 cm
Change in amplitude (= 1.5 – 1.3) = 0.2 cm
{OR Method 2 (more direct – based on graph): - this was the expected method to use
Change in energy = 0.60 mJ
We can use this fact to find the change in amplitude that corresponds to the change in energy found above from the graph directly.
We can look in the graph for the amplitude that corresponds to an energy of 0.6 mJ.
From the graph, when the energy changes by an amount of 0.6 mJ, the corresponding max amplitude (for 0.6 mJ) is 1.3 cm.
Initial max amplitude = 1.5 cm
New max amplitude = 1.3 cm
Change in amplitude = 1.5 – 1.3 = 0.2 cm}
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