A student investigated the horizontal oscillations of a trolley between two springs, using the apparatus shown - Edexcel - A-Level Physics - Question 6 - 2023 - Paper 3

To determine the maximum velocity of the trolley, we can analyze the data logger output:

1. Read from Oscilloscope: Count the number of divisions for one complete oscillation on the x-axis of the graph.
2. Time-Base Setting: Since the time-base is set to 250 ms per division, we can find the period (T) of the oscillation:
   • For example, if the graph displays 4 divisions for one full cycle, the period is: $T = 4 ext{ div} imes 250 ext{ ms/div} = 1000 ext{ ms} = 1 ext{ s}$
3. Calculate Frequency: The frequency (f) is the reciprocal of the period: f = rac{1}{T} = 1 ext{ Hz}
4. Determining Maximum Velocity: The maximum velocity (v_max) can be calculated as: v_{max} = A imes rac{2 ext{π}}{T}
   • Where A is the amplitude. If the amplitude was measured at 6 cm (0.06 m):
   = 0.06 imes 2 ext{π} \ ext{v_{max}} \ ext{= 0.377 m/s}$$ Therefore, the maximum velocity of the trolley is approximately 0.377 m/s.

The shape of the graph that shows how amplitude (A) varies with frequency (f) can be explained as follows:

1. Amplitude and Frequency Relationship: As the frequency increases, the amplitude of oscillation initially increases until it reaches a maximum point. Beyond this frequency, further increases in frequency typically lead to a reduction in amplitude.
2. Resonance Effect: This behavior usually indicates the presence of resonance at a particular frequency, where the external force matches the natural frequency of the system (in this case, the trolley-spring system).
3. Energy Transfer: At resonance, the energy transfer from the vibrator to the trolley is maximized, causing larger oscillations. As the frequency continues to increase past this point, the amplitude decreases due to damping effects, and less energy is available for oscillation, resulting in reduced amplitudes.

To determine the effective spring constant (k) of the oscillating trolley system, we can use the formula: $k = 4 ext{π}^2 m f^2$ Where:

• m = mass of the trolley = 0.87 kg
• f = frequency (which we determined from part (b)). Assuming we found an oscillation frequency of 1 Hz:

1. Calculating k: $$= 4 imes (3.14159)^2 imes 0.87 \ = 4 imes 9.8676 imes 0.87 \ = 34.36 ext{ N/m}$$

Thus, the effective spring constant k is approximately 34.36 N/m.