Work, Energy, and Simple Machines · Lesson 8 of 13
Conservation of Mechanical Energy
“Potential energy and kinetic energy keep trading places without losing the total receipt.”
• Calculate mechanical energy as the sum of kinetic and potential energy. • Explain energy conversion during free fall and pendulum motion. • State the condition for conservation of mechanical energy. • Use conservation to find speed without tracing every intermediate step. • Include friction correctly as work that changes mechanical energy.
Hold a ball at a height and it has gravitational potential energy but no kinetic energy. Release it and the ball speeds up while losing height. Just before reaching the ground, almost all the original gravitational potential energy has become kinetic energy. The two forms change, but under ideal conditions their sum does not.
Mechanical energy is the sum of the kinetic energy and potential energy of an object or system.
A Freely Falling Object
Consider an object of mass m released from rest at height h. At the top, its velocity is zero, so K = 0. Its potential energy is U = mgh. The initial mechanical energy is therefore mgh.
At an intermediate point, let the object be at height h prime and have speed v. Its potential energy is mgh prime and its kinetic energy is one-half mv squared. During the fall, the decrease in potential energy equals the increase in kinetic energy. Their sum remains mgh.
Showing the Conversion Mathematically
If the object has been falling for time t, its speed is v = gt because it started from rest. The distance fallen is one-half gt squared, so its remaining height is h prime = h - one-half gt squared. Substituting these into kinetic and potential energy makes the exchange visible.
Adding the two expressions cancels the equal positive and negative terms involving time. The result is mgh, the same mechanical energy the object possessed before release. This calculation demonstrates that the increase in kinetic energy is exactly equal to the decrease in potential energy during ideal free fall.
Immediately before the object reaches the chosen zero level, h is zero. Potential energy is therefore zero, while kinetic energy has reached mgh. The mechanical energy remains equal to its initial value.
When an object moves under gravity without external forces such as friction or air resistance changing its mechanical energy, the sum of kinetic and potential energy remains constant.
Activity: Let Us Experiment
Set up a simple pendulum and place a sheet behind it. Draw a horizontal line through the height of the bob at one extreme position. Release the bob from that level without pushing. It moves fastest at the lowest point and rises toward almost the same level on the other side.
At an extreme point, the bob momentarily stops: kinetic energy is zero and gravitational potential energy is greatest. At the lowest point, the chosen potential energy is zero and kinetic energy is greatest. At the opposite extreme, kinetic energy returns to zero and potential energy rises again. A real pendulum reaches slightly smaller heights because friction at the support and air resistance transfer mechanical energy to other forms.
| Pendulum Position | Kinetic Energy | Potential Energy |
|---|---|---|
| Extreme position | Zero | Maximum |
| Moving downward | Increasing | Decreasing |
| Lowest position | Maximum | Minimum or chosen as zero |
| Moving upward | Decreasing | Increasing |
Finding Speed at the Bottom of a Slide
Problem
A child starts from rest at the top of a frictionless slide of vertical height h. Find the speed at the bottom.
- 1.At the top, kinetic energy is zero and potential energy is mgh.
- 2.At the bottom, choose potential energy as zero and write kinetic energy as one-half mv².
- 3.Conservation gives one-half mv² = mgh.
- 4.Cancel mass m from both sides.
- 5.Multiply by two: v² = 2gh.
- 6.Take the positive square root for speed: v = √(2gh).
- 7.Mass cancels, so children of different masses reach the same ideal speed from the same height.
- 8.Only vertical height appears, so the ideal final speed does not depend on the slide’s shape.
When Friction Does Work
Mechanical energy is not conserved for an object when an external resistive force does negative work on it. The work-energy relation for the mechanical energy account becomes work by the resistive force equals final mechanical energy minus initial mechanical energy. Total energy is still accounted for, but part of the initial mechanical energy is transferred to thermal energy, deformation or sound.
Problem
A 10000 kg truck moving at 72 km h⁻¹ enters a ramp inclined so that it rises 1 m for every 2 m travelled. Sand exerts a backward force of 50000 N. Find the minimum ramp distance needed to stop it using g = 10 m s⁻².
- 1.Convert speed: 72 km h⁻¹ × 5/18 = 20 m s⁻¹.
- 2.Initial kinetic energy = one-half × 10000 × 20² = 2000000 J.
- 3.Take initial potential energy as zero, so initial mechanical energy is 2000000 J.
- 4.Let distance along the ramp be d. The vertical height gained is d/2.
- 5.Final kinetic energy is zero.
- 6.Final potential energy = mg(d/2) = 10000 × 10 × d/2 = 50000d J.
- 7.Work done by sand = -50000d J.
- 8.Use work by sand = final mechanical energy - initial mechanical energy.
- 9.-50000d = 50000d - 2000000.
- 10.Therefore, 100000d = 2000000 and d = 20 m.
Problem
A ball is released from rest at height h. Show that its mechanical energy just before reaching the ground is mgh.
- 1.Initial mechanical energy at height h is mgh because initial kinetic energy is zero.
- 2.With no air resistance, mechanical energy remains constant.
- 3.At ground level, potential energy is chosen as zero.
- 4.Therefore, kinetic energy just before impact must be mgh.
- 5.Mechanical energy at that instant is mgh + 0 = mgh.
Roller-Coaster Reasoning
On a descending track, potential energy decreases and kinetic energy increases. On an ascending track, kinetic energy decreases while potential energy increases. If friction were absent, every later peak could reach the original release height. Real tracks have friction and air resistance, so mechanical energy is gradually transferred to thermal energy and sound, making later peaks lower.
Do not automatically set initial mechanical energy equal to final mechanical energy. First check whether friction, air resistance or another external force does work. If it does, include that work in the energy equation.
Quiz
Which description best matches Mechanical Energy?
Which description best matches Conservation of Mechanical Energy?
Which term matches this description: Mechanical energy is the sum of the kinetic energy and potential energy of an object or system.
Which term matches this description: When an object moves under gravity without external forces such as friction or air resistance changing its mechanical energy, the sum of kinetic and potential energy remains constant.
Which statement is a key takeaway from this lesson?
Practice Problems
- A ball falls from rest through 5 m. Find its speed using g = 10 m s⁻² and neglect air resistance.
- Describe kinetic and potential energy at three positions of a swinging pendulum.
- Explain why two frictionless slides of different shapes give the same final speed when their heights are equal.
- A resistive force does -300 J of work on a system whose initial mechanical energy is 900 J. Find its final mechanical energy.
Key Takeaways
• Mechanical energy is K + U. • During ideal free fall, potential energy changes into kinetic energy. • Mechanical energy remains constant when no external resistive force changes it. • Conservation can determine speed directly from height. • Friction must be included as work that changes mechanical energy.