Interactive physics simulator
Mechanical Energy
Explore the interaction of kinetic and potential energy. Track vector force transitions, power generation rates, and elastic bungee limits in three realistic simulator environments designed to demonstrate the mechanical energy equation.
Mechanical Energy Lab
Interact with loop curvatures, hydroelectric pump rates, and bungee stiffeners to monitor the exchanges between kinetic, potential, and dissipated heat energies.
Live Telemetry
- Payload Mass (m)
- 100 kg
- Loop Radius (R)
- 4.5 m
- Current Height (h)
- 0.0 m
- Car Speed (v)
- 0.0 m/s
- Normal Force (F_N)
- 0 N
- Potential Energy
- 0 J
- Kinetic Energy
- 0 J
- Reservoir Height
- 50 m
- Flow Rate (Q)
- 4.0 m³/s
- Efficiency (η)
- 80%
- Hydraulic Power
- 0 kW
- Electrical Power
- 0 kW
- Stored Volume
- 10000 m³
- Stored GPE
- 0 MJ
- Jumper Mass (m)
- 75 kg
- Free Length (L)
- 25 m
- Stretch (x)
- 0.0 m
- Tension Force
- 0 N
- Decel Force
- 1.0 g
- Peak Decel
- 1.0 g
- Kinetic Energy
- 0 J
What is Mechanical Energy?
In physics, mechanical energy is the energy associated with the motion and position of an object. It represents the sum of macroscopic kinetic energy (KE, the energy of motion) and potential energy (PE, the stored energy of configuration):
Where kinetic energy is defined as KE = (1)/(2)mv2, and potential energy can take several forms, including gravitational potential energy (PEg = mgh) and elastic potential energy (PEe = (1)/(2)kx2).
Conservation of Mechanical Energy
When a physical system is acted upon only by conservative forces, its total mechanical energy remains constant. Conservative forces, such as gravity and spring restoring forces, do work that depends only on the initial and final states of the system, not on the path taken.
This allows us to solve complex physical trajectories—like loop-the-loop roller coasters or bouncing pendulums—solely by comparing the energies at the start and end coordinates without calculating intermediate force variations.
Non-Conservative Forces and Dissipation
If a system is subjected to non-conservative forces like friction, air resistance, or drag, the mechanical energy is no longer conserved. These forces perform negative work on the moving object, converting mechanical energy into thermal energy (<span class=inline-formula>E<sub>therm</sub></span>):
The mechanical energy decreases, but the total energy of the closed system (including the dissipated heat) is preserved.
Solved Numerical Examples
A roller coaster car of mass m = 200 kg is released from rest at a height of h = 15 meters above the ground. It glides down a frictionless track into a circular loop of radius R = 5.0 meters. (a) Determine if the car will complete the loop. (b) Find the speed of the car at the top of the loop. Use g = 9.8 m/s².
View Step-by-Step Solution
- Identify the given values: mass m = 200 kg, initial height h = 15 m, loop radius R = 5.0 m, and gravity g = 9.8 m/s².
- Check the loop-the-loop completion condition: The minimum height required to complete a frictionless loop is h_min = 2.5 · R.
- Calculate h_min: h_min = 2.5 · 5.0 = 12.5 meters. Since the actual height h = 15 m is greater than 12.5 m (h > h_min), the car will successfully complete the loop without losing contact.
- Apply conservation of mechanical energy between the release point (initial) and the top of the loop (final): E_initial = E_final.
- Write the energy equation: PE_initial + KE_initial = PE_top + KE_top. Since the car starts from rest, KE_initial = 0.
- Substitute the formulas: m · g · h = m · g · (2R) + 1/2 · m · v_top² (the height at the top of the loop is 2R = 10 m).
- Divide by mass (m) to simplify: g · h = 2 · g · R + 1/2 · v_top².
- Solve for speed v_top: v_top = √(2 · g · (h - 2R)) = √(2 · 9.8 · (15 - 10)) = √(19.6 · 5) = √(98) ≈ 9.90 m/s.
A pumped-storage hydroelectric plant pumps Q = 4.0 m³/s of water from a lower reservoir to an upper reservoir at a height of h = 80 meters. The pumping system is 75% efficient. How much electrical power in Megawatts (MW) does the pump draw from the grid? Use g = 9.8 m/s² and water density ρ = 1000 kg/m³.
View Step-by-Step Solution
- Identify the values: water density ρ = 1000 kg/m³, flow rate Q = 4.0 m³/s, height h = 80 m, gravity g = 9.8 m/s², and pumping efficiency η = 0.75 (75%).
- Calculate the rate of Gravitational Potential Energy gained by the water (Hydraulic Power): P_hydraulic = ρ · Q · g · h.
- Substitute the values: P_hydraulic = 1000 · 4.0 · 9.8 · 80 = 3,136,000 Watts = 3.136 MW.
- Relate the electrical input power to the hydraulic output power using pumping efficiency: η = P_hydraulic / P_electrical.
- Solve for P_electrical: P_electrical = P_hydraulic / η.
- Substitute the values: P_electrical = 3.136 / 0.75 ≈ 4.18 Megawatts.
A bungee jumper of mass m = 70 kg jumps from a bridge. The unstretched length of the bungee cord is L = 20 meters, and its spring constant is k = 100 N/m. Assuming no air resistance and g = 9.8 m/s², calculate the maximum stretch distance (x) of the bungee cord and the jumper's maximum speed during the fall.
View Step-by-Step Solution
- Identify the values: mass m = 70 kg, unstretched length L = 20 m, spring constant k = 100 N/m, and gravity g = 9.8 m/s².
- Apply conservation of energy between the bridge (y = 0, initial) and the lowest point where speed is momentarily zero (final): E_initial = E_final.
- Set the reference height y = 0 at the bridge. Initial mechanical energy E_initial = 0. Final height is -(L + x), where x is the cord stretch.
- Write the final energy equation: PE_grav + PE_elastic + KE = m · g · (-(L + x)) + 1/2 · k · x² + 0 = 0.
- Rearrange into a quadratic equation: 1/2 · k · x² - m · g · x - m · g · L = 0. Substitute values: 50 · x² - 686 · x - 13720 = 0.
- Simplify: x² - 13.72 · x - 274.4 = 0. Solve using quadratic formula: x = (13.72 + √(13.72² - 4 · 1 · (-274.4))) / 2 = (13.72 + √(188.24 + 1097.6)) / 2 = (13.72 + √(1285.84)) / 2 ≈ (13.72 + 35.86) / 2 ≈ 24.79 meters.
- To find the maximum speed, it occurs where net force is zero: F_net = 0 ⇒ m · g = k · x_eq ⇒ x_eq = m · g / k = 70 · 9.8 / 100 = 6.86 meters.
- Apply conservation of energy from bridge to x_eq: PE_initial = PE_grav_eq + PE_elastic_eq + KE_max ⇒ 0 = -m · g · (L + x_eq) + 1/2 · k · x_eq² + 1/2 · m · v_max².
- Substitute values: 0 = -70 · 9.8 · (20 + 6.86) + 0.5 · 100 · 6.86² + 35 · v_max² ⇒ 0 = -18425.68 + 2352.98 + 35 · v_max² ⇒ 35 · v_max² = 16072.7 ⇒ v_max = √(459.22) ≈ 21.43 m/s.
Conceptual Practice
Why is the minimum drop height to complete a loop-the-loop larger than the loop height itself? Explain using potential and kinetic energy.
Show Explanation
At the top of the loop, the car cannot have zero speed; it must have a minimum centripetal speed to stay on the track. This speed is v_min = √(gR). At the top of the loop (height h = 2R), the car must have both gravitational potential energy (mg · 2R) and kinetic energy (1/2 · m · v_min² = 1/2 · m · gR). Summing these gives a total energy of 2.5 · mgR. Since the initial energy is purely potential energy (mgh), the minimum starting height must be h = 2.5R, which is higher than the loop diameter 2R.
Explain how friction affects the total mechanical energy versus the total energy of a roller coaster system.
Show Explanation
Friction is a non-conservative force. It does negative work on the roller coaster, converting a portion of its kinetic and potential energy into thermal energy. Consequently, the total mechanical energy (E_mech = KE + PE) decreases over time. However, the total energy of the universe (mechanical + thermal energy) remains strictly conserved, in accordance with the first law of thermodynamics.
In a pumped-storage hydroelectric system, why is the round-trip efficiency always less than 100%? Name three sources of energy dissipation.
Show Explanation
The round-trip efficiency is less than 100% because mechanical energy is converted into non-recoverable thermal energy by friction and resistance during both pumping and generation. Three sources of energy loss are: (1) fluid friction (viscosity) as water flows through the penstock pipe, (2) electromagnetic drag and heat loss in the generator/motor coils, and (3) mechanical friction in the turbine bearings.
During a bungee jump, at what point during the fall does the jumper reach their maximum speed? Is it right before the cord starts stretching?
Show Explanation
No, the maximum speed does not occur when the cord starts stretching. When the cord starts to stretch (y_fall = L), the bungee tension force is zero, and gravity is still the only force accelerating the jumper downward. The jumper continues to speed up until the elastic upward force equals the downward gravitational force (F_net = 0, or kx = mg). Beyond this equilibrium point, tension exceeds gravity, causing the jumper to decelerate. Thus, maximum speed is reached when the cord has stretched by x = mg/k.
Frequently Asked Questions
What is mechanical energy?
Mechanical energy (Emech) is the sum of kinetic energy (KE, the energy of motion) and potential energy (PE, the stored energy of position or configuration) in a physical system: Emech = KE + PE.
What is the Principle of Conservation of Mechanical Energy?
The Principle of Conservation of Mechanical Energy states that if only conservative forces (like gravity or spring forces) do work on a system, the total mechanical energy of that system remains constant: Ei = Ef, or KEi + PEi = KEf + PEf.
What is the difference between mechanical energy and total energy?
Mechanical energy only includes macroscopic kinetic energy and potential energy. Total energy includes all forms, such as thermal, chemical, electrical, and nuclear energy. While total energy is always conserved in any process, mechanical energy can decrease if non-conservative forces like friction convert it into thermal energy.
How do conservative and non-conservative forces differ?
A conservative force (like gravity or a spring) does work that is independent of the path taken, allowing mechanical energy to be fully stored and recovered. A non-conservative force (like friction or air resistance) does work that depends on the path, converting mechanical energy into non-recoverable forms like heat.
How does a roller coaster loop demonstrate mechanical energy conservation?
At the starting high point, the coaster has maximum gravitational potential energy (PEg = mgh) and zero kinetic energy. As it descends, PEg converts to KE. As it enters the circular loop, some KE converts back to PEg to climb the loop. At the top of the loop, it has both PEg and KE.
What is the minimum height needed to complete a loop-the-loop?
In a frictionless loop of radius R, the minimum height from which a car must start from rest to complete the loop without losing contact is hmin = 2.5 R. At this height, the speed of the car at the top is exactly v = √(g R), resulting in the normal force FN dropping to exactly zero at the top.
What happens if a roller coaster car enters a loop-the-loop below the critical height?
If the car starts below 2.5 R, its speed at the top of the loop will fall below √(g R). The required centripetal force will exceed the gravitational force alone. Since gravity is the only downward force and the track can only push away (normal force FN ≥ 0), the car will lose contact with the track and fall off, following a parabolic projectile path.
How does pumped storage hydroelectricity store mechanical energy?
During low demand, surplus electricity powers water pumps that lift water from a lower reservoir to an upper reservoir, converting electrical energy into gravitational potential energy. During peak demand, the water is released down through turbines, converting GPE back into kinetic energy to generate electricity.
How is the efficiency of a hydroelectric pump-generator calculated?
Efficiency (η) is the ratio of useful energy output to total energy input. For generator mode, η = Pelectrical / (rho Q g h). For pump mode, η = (rho Q g h) / Pelectrical. The overall round-trip efficiency of modern pumped-storage plants is typically 70% to 80%.
How does energy transform during a bungee jump?
Before jumping, the jumper has maximum GPE. During the free fall, GPE converts to KE. Once the bungee cord reaches its unstretched length and begins to stretch, both GPE and KE convert into elastic potential energy (PEe = 1/2 k x2) until the jumper stops momentarily at the lowest point, where energy is stored as elastic potential energy.
What roles do air resistance and internal cord friction play in bungee jumping?
They act as non-conservative forces, doing negative work on the system. They dissipate mechanical energy into thermal energy, causing the peak jump height and rebound height to decrease with each oscillation until the jumper comes to rest.
How does the spring constant <span class=inline-formula>k</span> affect bungee jump dynamics?
A larger spring constant k (stiffer cord) exerts a stronger restoring force for the same stretch distance, causing the jumper to stop in a shorter distance, experience higher deceleration G-forces, and bounce back more rapidly.