Free Fall Method Experiment

Determination of Acceleration Due to Gravity by Free Fall Method

1. Aim

To determine the acceleration due to gravity (g) by measuring the time of free fall of an object through a known height and to verify the kinematic equations of motion under constant acceleration.

2. Apparatus Used

  • Free fall apparatus (electromagnet with release mechanism)
  • Steel ball
  • Electronic timer with microsecond resolution
  • Vertical stand with measuring scale (least count 1mm)
  • Spirit level
  • Trap door or impact sensor (optional)
  • Plumb line

3. Diagram

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4. Theory

When an object is released from rest and allowed to fall freely under gravity, its motion is described by the kinematic equations for uniformly accelerated motion:

\[ h = \frac{1}{2}gt^2 \]

Where:

  • \( h \) = Vertical distance fallen
  • \( g \) = Acceleration due to gravity
  • \( t \) = Time of fall

Rearranging the equation gives:

\[ g = \frac{2h}{t^2} \]

This experiment measures the time (t) taken for an object to fall through a measured height (h), from which g can be calculated.

Note: Air resistance is neglected in this experiment, which is valid for dense objects (like steel balls) falling moderate distances. For more precise measurements, corrections for air resistance may be applied.

5. Formula

The acceleration due to gravity is calculated using:

\[ g = \frac{2h}{t^2} \]

For multiple measurements, a more accurate method is to plot \( h \) vs \( t^2 \) graph:

\[ \text{Slope} = \frac{h}{t^2} = \frac{g}{2} \] \[ \therefore g = 2 \times \text{Slope} \]

Percentage error calculation:

\[ \% \text{Error} = \left| \frac{g_{\text{experimental}} - g_{\text{standard}}}{g_{\text{standard}}} \right| \times 100 \]

6. Procedure

  1. Set up the free fall apparatus vertically using a plumb line and spirit level.
  2. Measure the height (h) from the release point to the impact sensor/trap door using the scale.
  3. Place the steel ball on the electromagnet release mechanism.
  4. Set the electronic timer to zero and ensure proper triggering.
  5. Release the ball by switching off the electromagnet and record the fall time (t).
  6. Repeat the measurement for the same height at least 5 times.
  7. Repeat steps 2-6 for different heights (e.g., 50cm, 75cm, 100cm, 125cm, 150cm).
  8. Record all observations in the tabular form.
  9. Calculate g for each height and take average.
  10. Plot graph of h vs t² and determine g from the slope.

Important: The timer should start exactly when the ball is released and stop exactly when it hits the sensor. Modern setups use light gates or magnetic triggers for precise timing.

7. Observation Table

S.No. Height (h) in cm Time of fall (t) in seconds Mean time (tavg) tavg2 (s²) g = 2h/t² (m/s²)
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5
1 50.0
2 75.0
3 100.0
4 125.0
5 150.0

8. Calculations

  1. For each height, calculate mean time of fall:
    \[ t_{avg} = \frac{t_1 + t_2 + t_3 + t_4 + t_5}{5} \]
  2. Calculate \( t_{avg}^2 \) for each height.
  3. Calculate g for each height:
    \[ g = \frac{2h}{t_{avg}^2} \] (convert h to meters)
  4. Calculate average value of g from all measurements.
  5. Plot graph of h (y-axis) vs t² (x-axis):
    • The graph should be a straight line passing through origin
    • Determine slope (h/t²)
    • Calculate \( g = 2 \times \text{slope} \)
  6. Compare with standard value (9.81 m/s²) and calculate percentage error.

Graph Plotting

Expected Graph: A straight line through origin with slope = g/2

Procedure:

  1. Plot height h (in meters) on y-axis
  2. Plot \( t_{avg}^2 \) on x-axis
  3. Draw best-fit straight line
  4. Calculate slope (Δh/Δt²)
  5. Determine g = 2 × slope

9. Result

  • The acceleration due to gravity (g) determined from the experiment:
    • From individual calculations = _____ ± _____ m/s²
    • From graph slope = _____ m/s²
  • Standard value of g at the location = 9.81 m/s²
  • Percentage error = _____ %
  • The graph of h vs t² is a straight line through origin, verifying the equation \( h = \frac{1}{2}gt^2 \)

10. Precautions

  • Ensure the apparatus is perfectly vertical using plumb line/spirit level
  • Measure heights accurately from release point to impact point
  • Use precise electronic timing system with microsecond resolution
  • Ensure clean release of the ball without initial velocity
  • Repeat measurements multiple times for each height
  • Use a dense, spherical object to minimize air resistance effects
  • Perform experiment in a draft-free environment
  • Check that the timing mechanism triggers precisely at release and impact
  • Account for any delay in the timing mechanism if known

11. Viva Voce Questions

1. What is free fall? What forces act on an object in free fall near Earth's surface?
2. Derive the equation \( h = \frac{1}{2}gt^2 \) from basic kinematic equations.
3. Why do we neglect air resistance in this experiment? When would this approximation fail?
4. How does the value of g vary with height above Earth's surface?
5. What would be the effect on your results if the apparatus wasn't perfectly vertical?
6. Why do we plot h vs t² instead of h vs t? What does the slope represent?
7. How would your results change if the object had an initial velocity when released?
8. What are the advantages of the free fall method compared to pendulum methods for determining g?
9. How could you modify this experiment to measure g more accurately?
10. What would be the shape of the h vs t graph? Why?
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