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Mechanical Energy Conservation: Free Fall Guide

PhysicsIntermediateReading time: 3 min

Overview

Verify the law of conservation of mechanical energy by observing the free fall of a heavy object. Confirm that the decrease in gravitational potential energy equals the increase in kinetic energy when only gravity does work.

Background

The idea of conservation of mechanical energy dates back to Galileo's study of pendulum motion, where he found that the bob always rises to the same height as its release point. Later, Leibniz proposed the concept of conservation of 'vis viva' (living force, i.e., mv2mv^2). Eventually, Joule and others established the complete law of conservation of energy. This experiment uses a ticker timer to record free fall motion and quantitatively verify mechanical energy conservation.

Key Concepts

Gravitational Potential Energy (EpE_p)

Ep=mghE_p = mgh

The energy possessed by an object due to its vertical position, proportional to height.

Kinetic Energy (EkE_k)

Ek=12mv2E_k = \frac{1}{2}mv^2

The energy possessed by an object due to its motion, proportional to the square of its velocity.

Mechanical Energy (EE)

E=Ep+EkE = E_p + E_k

The sum of kinetic and potential energy in a system.

Formulas & Derivation

Law of Conservation of Mechanical Energy

ΔEp=ΔEk⇒mgh=12mv2\Delta E_p = \Delta E_k \Rightarrow mgh = \frac{1}{2}mv^2
If the initial velocity is 0, the potential energy lost after falling height hh equals the kinetic energy gained.

Instantaneous Velocity Calculation

vn=hn+1−hn−12Tv_n = \frac{h_{n+1} - h_{n-1}}{2T}
For an object in uniform acceleration, the average velocity over a time interval equals the instantaneous velocity at the midpoint of that interval.

Experiment Steps

  1. 1

    Apparatus Adjustment

    Ensure the ticker timer is fixed vertically and the limiting holes are aligned vertically to minimize tape friction. Connect the power supply (connected by default in simulation).
  2. 2

    Tape Operation

    Turn on the timer power first (stabilize ticking), then release the heavy bob. Observe if the bob falls smoothly.
  3. 3

    Trace Selection

    Find a clear first dot (t=0t=0). If the distance between the 1st and 2nd dots is close to 2mm2mm (at 50Hz50Hz frequency), the initial velocity can be considered 0.
  4. 4

    Measurement and Calculation

    Select widely spaced counting points (e.g., take 1 counting point every 5 dots, so T=0.1sT=0.1s). Measure the fall height hh for each point and calculate the corresponding instantaneous velocity vv.

Learning Outcomes

  • Verified that mechanical energy is conserved during free fall within the range of experimental error.
  • Mastered the method of using 'mid-time instantaneous velocity' to process ticker tape data.
  • Analyzed the causes of systematic errors (decrease in EpE_p is slightly larger than increase in EkE_k) due to air resistance and tape friction.

Real-world Applications

  • Hydroelectric Power: Water's gravitational potential energy is converted into kinetic energy by a dam, which then drives turbines to generate electricity.
  • Pile Driver: A heavy hammer is lifted to gain potential energy, which converts to massive kinetic energy upon falling to drive piles into the ground.
  • Roller Coaster: Trains constantly convert between kinetic and potential energy as they move up and down the tracks.

Common Misconceptions

Misconception
The falling speed of an object depends on its mass; heavier objects fall faster.
Correct
Free fall acceleration gg is independent of mass. The rate of fall is only affected by air resistance (ignored in this experiment).
Misconception
Release the tape first, then turn on the power.
Correct
Power should be turned on first to stabilize ticking before releasing the tape. Otherwise, the beginning of the tape may be blank or dots may be unstable.
Misconception
Velocity can be calculated using v=gtv = gt or v=2ghv = \sqrt{2gh}.
Correct
In a verification experiment, you cannot use the formula of the law being verified to calculate data. You must use the average velocity of the tape v=stv = \frac{s}{t} to measure instantaneous velocity.

Further Reading

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