Sp Unit 3.2
Practicals
Vibrations
SP Unit 3.2PracticalsVibrationsLearners should be able to demonstrate and apply their knowledge and understanding of: |
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|---|---|
| 1. | Measurement of g with a pendulum |
| 2. | Investigation of the damping of a spring |
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1. Measurement of g with a Pendulum
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⇒ Objective:
- To determine the acceleration due to gravity (g) using a simple pendulum.
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⇒ Apparatus Required:
- – A small metal bob
- – A long inextensible thread (~1 meter)
- – A rigid support with a clamp
- – A stopwatch
- – A meter ruler
- Figure 1 Simple pendulum
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⇒ Theory:
- A simple pendulum consists of a small bob suspended from a fixed point by a light inextensible string. When displaced slightly and released, it undergoes simple harmonic motion with a period (T) given by:
- [math]T = 2\pi \sqrt{\frac{L}{g}}[/math]
- Where:
- – T = time period of the pendulum (s),
- – L = length of the pendulum (m),
- – g = acceleration due to gravity (m/s²).
- Rearranging for g:
- [math]g = \frac{4\pi^2 L}{T^2}[/math]
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⇒ Procedure:
- 1. Setting Up the Pendulum:
- – Attach a small bob to one end of a thread and fix the other end to a rigid support.
- – Measure the length (L) from the point of suspension to the center of the bob using a meter ruler.
- 2. Performing the Experiment:
- – Displace the pendulum slightly (small angle <[math]15^o[/math]) and release it.
- – Start the stopwatch when the pendulum passes the central position.
- – Record the time for 10 complete oscillations.
- – Calculate the time period (T) as:
- [math]T = \frac{\text{Total time for 10 oscillations}}{10}[/math]
- 3. Repeating the Experiment:
- – Vary the length (L) of the pendulum and repeat the measurements.
- – Plot a graph of [math]T^2[/math] (y-axis) vs. L (x-axis).
- – The slope of the graph will be [math]\frac{4\pi^2}{g}[/math], from which g can be determined.
- Figure 2 Graph between t and l while the second graph between [math]T^2[/math] and L
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⇒ Observations & Data Table:
| Length L (m) | Time for 10 oscillations (s) | Time period T (s) | [math]T^2[/math](s²) |
|---|---|---|---|
| 0.50 | 14.2 | 1.42 | 2.02 |
| 0.75 | 17.3 | 1.73 | 2.99 |
| 1.00 | 20.1 | 2.01 | 4.04 |
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⇒ Result:
- The experimental value of g should be close to [math]9.81m/s^2[/math], with minor errors due to air resistance and measurement inaccuracies.
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⇒ Precautions:
- – Ensure the pendulum swings in a single plane.
- – Measure the length accurately from the fixed point to the center of the bob.
- – Use a small angle (< [math]15^o[/math]) to maintain simple harmonic motion.
- – Start and stop the stopwatch carefully to reduce reaction time error.
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⇒ Conclusion:
- By measuring the time period for different lengths of the pendulum, we determined the acceleration due to gravity (g) experimentally. The results should be close to the standard value of [math]9.81m/s^2[/math].
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2. Investigation of the Damping of a Spring
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⇒ Objective:
- To study how the amplitude of oscillations of a spring-mass system decreases over time due to damping forces.
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⇒ Apparatus Required:
- – A helical spring
- – A mass hanger with slotted weights
- – A stopwatch
- – A meter ruler
- – A damping medium (e.g., water, oil, or air resistance)
- Figure 3 Damping oscillation
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⇒ Theory:
- A mass attached to a spring oscillates with simple harmonic motion. However, due to resistive forces (air resistance, friction, or a damping medium), the amplitude of oscillations gradually decreases over time. The equation governing damped motion is:
- [math]x(t) = A e^{-\gamma t} \cos(\omega t)[/math]
- Where:
- – x(t) is the displacement at time t,
- – A is the initial amplitude,
- – γ is the damping coefficient,
- – ω is the angular frequency of oscillations.
- Damping can be categorized into three types:
- Underdamping (gradual decrease in amplitude).
- Critical damping (oscillation stops in the shortest time).
- Overdamping (system returns to equilibrium without oscillating).
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⇒ Procedure:
- 1. Setting Up the Experiment:
- – Suspend a spring vertically from a fixed support.
- – Attach a known mass to the free end and allow it to come to rest.
- – Pull the mass downward slightly and release it to start oscillations.
- 2. Measuring Damping:
- – Record the amplitude of oscillations after each complete cycle.
- – Repeat the experiment with different damping conditions (air, water, oil).
- – Plot a graph of amplitude vs. time to observe the rate of damping (as shown above).
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⇒ Observations & Data Table:
| Time (s) | Amplitude (cm) (Air) | Amplitude (cm) (Water) | Amplitude (cm) (Oil) |
|---|---|---|---|
| 0 | 10 | 10 | 10 |
| 5 | 8.5 | 7.2 | 4.5 |
| 10 | 7.0 | 4.8 | 2.1 |
| 15 | 5.5 | 2.5 | 0.8 |
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⇒ Result:
- – The damping effect is least in air, higher in water, and maximum in oil.
- – The oscillations decay exponentially with time.
- – In a highly damped system (oil), the mass returns to equilibrium without oscillating (overdamping).
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⇒ Precautions:
- – Ensure the mass is not swinging sideways.
- – Use an appropriate damping medium for clear observation.
- – Measure amplitudes precisely using a ruler or video analysis.
- – Minimize external disturbances that could affect oscillations.
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⇒ Conclusion:
- The damping effect on a spring-mass system was studied by observing how the amplitude of oscillations decreases over time in different damping conditions. The experiment demonstrated the principles of underdamping, critical damping, and overdamping.