Interactive physics simulator
Conservation of Energy
Investigate the fundamental law of nature. Experience the conversion of gravitational potential energy, kinetic energy, elastic potential energy, and dissipated thermal loss in real time across three interactive lab models.
Conservation of Energy Lab
Interact with mass, height, gravity, friction, and spring constants to study the conservation and dissipation of mechanical energy.
Live Telemetry
- Payload Mass (m)
- 80 kg
- Current Height (h)
- 10.0 m
- Coaster Speed (v)
- 0.0 m/s
- Potential Energy (PE)
- 0 J
- Kinetic Energy (KE)
- 0 J
- Thermal Loss (E_th)
- 0 J
- Total Energy (E)
- 0 J
- Bob Mass (m)
- 2.0 kg
- String Length (L)
- 5.0 m
- Current Angle (θ)
- 45°
- Linear Speed (v)
- 0.0 m/s
- Potential Energy (PE)
- 0 J
- Kinetic Energy (KE)
- 0 J
- Total Energy (E)
- 0 J
- Block Mass (m)
- 1.5 kg
- Stiffness (k)
- 400 N/m
- Compression (x)
- 1.00 m
- Block Speed (v)
- 0.0 m/s
- Elastic PE (EPE)
- 0 J
- Grav. PE (GPE)
- 0 J
- Kinetic Energy (KE)
- 0 J
The Law of Conservation of Energy
The Law of Conservation of Energy is a fundamental principle of physics stating that the total amount of energy in an isolated system remains constant. Energy can neither be created nor destroyed; it can only be transformed from one state into another.
In mechanical systems, energy typically oscillates between kinetic energy (energy of motion) and potential energy (stored energy of position). In the absence of friction and other non-conservative forces, the total mechanical energy (<span class=inline-formula>E<sub>mech</sub></span>) is conserved:
Where:
- KE is Kinetic Energy (1/2 · m · v²)
- PE is Potential Energy (such as Gravitational mgh or Elastic 1/2 · kx²)
Dissipation: Friction and Thermal Energy
In real-world systems, non-conservative forces like friction, air resistance, and viscosity act on moving bodies. These forces do negative work, transferring mechanical energy out of the system.
According to the conservation of total energy, this mechanical energy is not lost to the universe; rather, it is dissipated as thermal energy (heat) in the object and its surroundings. The expanded equation accounting for friction is:
In our Rollercoaster Lab and Pendulum Swing Lab, the gradual decay in speed and amplitude is caused by this heat conversion. The mechanical energy decreases, but the sum of mechanical and thermal energy remains strictly constant.
Solved Numerical Examples
A roller coaster car of mass m = 250 kg starts from rest at the top of a hill at a height of h₁ = 18 meters. It rolls down a frictionless track to a lower point at a height of h₂ = 4.0 meters. Calculate the car's speed at this lower point. Use g = 9.8 m/s².
View Step-by-Step Solution
- Identify the given values: mass m = 250 kg, initial height h₁ = 18 m, final height h₂ = 4.0 m, and gravity g = 9.8 m/s².
- State the conservation of mechanical energy principle: E_initial = E_final. Since there is no friction, the total mechanical energy is constant.
- Write the energy equation: PE₁ + KE₁ = PE₂ + KE₂.
- Substitute the definitions: m · g · h₁ + 1/2 · m · v₁² = m · g · h₂ + 1/2 · m · v₂².
- Since the car starts from rest, the initial velocity v₁ = 0, which simplifies the equation to: m · g · h₁ = m · g · h₂ + 1/2 · m · v₂².
- Divide all terms by mass (m) to eliminate it: g · h₁ = g · h₂ + 1/2 · v₂².
- Rearrange to solve for v₂: 1/2 · v₂² = g · (h₁ - h₂) ⇒ v₂ = √(2 · g · (h₁ - h₂)).
- Substitute the values: v₂ = √(2 · 9.8 · (18 - 4.0)) = √(19.6 · 14.0) = √(274.4) ≈ 16.57 m/s.
A simple pendulum consists of a 1.2 kg bob suspended by a string of length L = 2.5 meters. The bob is pulled aside to an initial angle of θ₀ = 60° and released from rest. What is the speed of the bob at the lowest point of its swing? (Assume no air resistance and g = 9.8 m/s²).
View Step-by-Step Solution
- Identify the values: mass m = 1.2 kg, length L = 2.5 m, release angle θ₀ = 60°, and gravity g = 9.8 m/s².
- Determine the initial height (h₀) of the bob relative to its lowest equilibrium position: h₀ = L - L · cos(θ₀) = L · (1 - cos(θ₀)).
- Calculate h₀: h₀ = 2.5 · (1 - cos(60°)) = 2.5 · (1 - 0.5) = 1.25 meters.
- Apply conservation of mechanical energy: PE_initial + KE_initial = PE_final + KE_final.
- At release point (initial): KE_initial = 0. At the lowest point (final): h_final = 0, so PE_final = 0. Therefore: PE_initial = KE_final.
- Write the simplified equation: m · g · h₀ = 1/2 · m · v².
- Divide by mass (m) and solve for speed: v = √(2 · g · h₀).
- Substitute the values: v = √(2 · 9.8 · 1.25) = √(24.5) ≈ 4.95 m/s.
A 0.80 kg wooden block is compressed against a horizontal spring with spring constant k = 500 N/m. The spring is compressed by x = 0.30 meters and released. The block is launched up a frictionless ramp inclined at 30°. Calculate the maximum vertical height (h) and the maximum distance along the ramp (d) that the block travels before stopping. (Use g = 9.8 m/s²).
View Step-by-Step Solution
- Identify given parameters: block mass m = 0.80 kg, spring constant k = 500 N/m, compression x = 0.30 m, incline angle θ = 30°, and gravity g = 9.8 m/s².
- Calculate the initial energy stored in the spring as Elastic Potential Energy: EPE_initial = 1/2 · k · x² = 0.5 · 500 · (0.30)² = 22.5 Joules.
- Apply conservation of energy: Since the ramp is frictionless, the initial EPE converts fully into Gravitational Potential Energy (GPE) at the peak height: E_initial = E_final ⇒ EPE_initial = GPE_final.
- Write the equation for peak height: 1/2 · k · x² = m · g · h.
- Solve for vertical height h: h = (1/2 · k · x²) / (m · g).
- Substitute values: h = 22.5 / (0.80 · 9.8) = 22.5 / 7.84 ≈ 2.87 meters.
- Determine the distance along the ramp (d) using trigonometry: sin(30°) = height / distance = h / d.
- Solve for distance d: d = h / sin(30°) = 2.87 / 0.5 = 5.74 meters.
Conceptual Practice
What is the difference between conservative and non-conservative forces in the context of the conservation of energy? Give examples of each.
Show Explanation
Conservative forces (like gravity and spring force) conserve mechanical energy. The work they do depends only on initial and final positions, not on the path taken. The energy can be fully recovered. Non-conservative forces (like friction, air resistance, and tension) dissipate mechanical energy. The work they do depends on the path, converting mechanical energy into non-recoverable thermal energy or internal energy.
A skater rolls down a U-ramp. If we double the skater's mass, how does the maximum speed at the bottom of the ramp change? Explain using energy principles.
Show Explanation
The maximum speed does not change. By conservation of energy, PE_top = KE_bottom ⇒ mgh = 1/2 · mv². Since mass (m) is on both sides of the equation, it cancels out: v = √(2gh). Therefore, the speed depends only on gravity and height, not on the mass of the skater.
A toy gun launches a small plastic ball vertically using a compressed spring. If the spring constant is doubled while keeping the compression distance constant, what happens to the maximum height reached by the ball?
Show Explanation
The maximum height is doubled. The initial energy is stored as elastic potential energy: E_initial = 1/2 · kx². At the peak height, it is all gravitational potential energy: E_final = mgh. Under conservation of energy, mgh = 1/2 · kx² ⇒ h = (k · x²) / (2mg). Since height h is directly proportional to spring constant k (h ∝ k), doubling k doubles the maximum height h.
A grandfather clock's pendulum swings back and forth. Why does it eventually stop swinging if it is not driven by an internal weight or spring? Where does the initial energy go?
Show Explanation
In a real pendulum, non-conservative forces like air resistance and pivot friction do negative work on the system. Over time, this work dissipates mechanical energy (KE + GPE), converting it into thermal energy in the surrounding air and pivot. This causes the amplitude to decay until the pendulum stops. To keep it swinging, clock mechanisms must continuously supply tiny amounts of mechanical energy.
Frequently Asked Questions
What is the Law of Conservation of Energy?
The Law of Conservation of Energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant.
What is mechanical energy?
Mechanical energy (Emech) is the sum of kinetic energy (KE) and potential energy (PE) in a physical system: Emech = KE + PE.
What is the difference between open, closed, and isolated systems?
An open system can exchange both energy and matter with its surroundings. A closed system can exchange energy, but not matter. An isolated system can exchange neither energy nor matter with its surroundings; its total energy is strictly constant.
How does friction affect the conservation of energy?
Friction is a non-conservative force. It does not destroy energy; rather, it transforms mechanical energy (KE and PE) into thermal energy (heat). While mechanical energy decreases, the total energy of the system plus its environment remains constant.
Why does a bouncing ball not reach its original height?
When a ball bounces, it deforms upon impact. Internal friction (hysteresis) and the impact force convert some of its kinetic energy into thermal energy and sound. Since it loses mechanical energy during the bounce, it rebounds with less kinetic energy, reaching a lower peak height.
What is a conservative force?
A conservative force is one for which the work done in moving an object between two points is independent of the path taken. Examples include gravitational force, electrostatic force, and spring force.
What is the SI unit of energy?
The SI unit of energy is the Joule (J). One Joule is defined as the work done by a force of one Newton moving an object through a distance of one meter (1 J = 1 N·m = 1 kg·m2/s2).
Can energy be converted with 100% efficiency?
Mechanical energy can be converted to thermal energy with 100% efficiency. However, converting thermal energy back into mechanical work is limited by the laws of thermodynamics (specifically the Carnot limit), meaning it can never be 100% efficient in a cyclic process.
How does a pendulum demonstrate conservation of energy?
At the highest point of its swing, the pendulum is momentarily at rest, so its energy is 100% gravitational potential energy (PE = mgh). As it swings downward, height decreases and speed increases, converting PE to kinetic energy. At the bottom, its height is zero, meaning its energy is 100% kinetic energy.
What is the formula for the total energy of a spring-mass vertical launcher?
The total energy is the sum of three terms: E = KE + GPE + EPE = (1)/(2)mv2 + mgh + (1)/(2)kx2, where x is the spring displacement from equilibrium and h is the height.
Does the conservation of energy apply to chemical and nuclear reactions?
Yes. In chemical reactions, energy is exchanged between kinetic energy and chemical potential energy. In nuclear reactions, mass-energy equivalence (E = mc2) applies: mass defect is converted into kinetic energy of products and high-energy photons, conserving total relativistic mass-energy.
Who discovered the conservation of energy?
The principle was developed in the mid-19th century by several scientists, including Julius Robert von Mayer, James Prescott Joule, and Hermann von Helmholtz, who formalized the mathematical equivalence of heat, work, and mechanical energy.