Interactive physics simulator
Potential Energy: PE = mgh
Investigate gravitational potential energy. Explore mechanical work against gravity, analyze skateboarding energy conversion along U-shaped ramps, and drop heavy weights with a pile driver to solve soil resistance.
Potential Energy Lab
Interact with mass, height, gravity, friction, and soil stiffness to analyze gravitational potential energy scaling and work.
Live Telemetry
- Object Mass (m)
- 2.0 kg
- Height (h)
- 5.0 m
- Gravity (g)
- 9.8 m/s²
- Gravity Force (F_g)
- 19.6 N
- Tension Force (F_T)
- 19.6 N
- Work Done (W)
- 98.0 J
- Potential Energy (PE)
- 98.0 J
- Mass (m)
- 50 kg
- Current Height (h)
- 8.0 m
- Current Speed (v)
- 0.0 m/s
- Potential Energy (PE)
- 3920 J
- Kinetic Energy (KE)
- 0 J
- Thermal Loss (E_th)
- 0 J
- Total Energy (E)
- 3920 J
- Hammer Mass (M)
- 100 kg
- Drop Height (H)
- 4.0 m
- Impact Speed (v_imp)
- 8.8 m/s
- Available PE (PE₀)
- 3920 J
- Penetration (d)
- 0.00 m
- Soil Resist. Force
- 0 kN
- Avg Soil Resist.
- 0 N
What is Potential Energy?
Potential Energy (PE) is the stored energy that an object possesses because of its position, shape, or configuration relative to other objects. It represents work done on a body to configure it into a state of potential work release. In a gravitational system, potential energy is stored due to the gravitational attraction between the object and the Earth.
The standard equation calculating gravitational potential energy near a planet's surface is:
Where:
- m is the mass of the object in kilograms (kg)
- g is the local acceleration due to gravity (approximately 9.8 m/s² on Earth's surface)
- h is the vertical elevation or height above a chosen reference level (m)
Deriving PE = mgh from Work
The potential energy formula is directly derived from the definition of mechanical work (W = F · s). When you lift an object of mass m vertically upward at a constant speed:
- You must exert an upward force (Flift) equal and opposite to the object's downward weight (Fg = mg).
- The displacement is the vertical height h (s = h).
- Since the lifting force acts in the direction of motion, the work done by the lifting force is:
W = Flift · s = (mg) · h = mgh
- According to the Work-Energy Theorem, this work is not lost; it is stored in the gravitational field as potential energy, meaning:
Δ PE = mgh
Conservation of Mechanical Energy
Gravitational potential energy acts as a temporary reservoir of energy. In conservative systems (where resistive forces like friction and air resistance are negligible), potential energy can be fully converted into kinetic energy and vice versa.
As a skater descends a U-shaped half-pipe, height decreases (losing PE), but speed increases (gaining KE). At the bottom of the ramp, potential energy is zero, and kinetic energy is at its maximum, where:
This relation demonstrates that the final speed of a falling object in a frictionless system is independent of its mass, relying solely on gravity and drop height.
Solved Numerical Examples
A construction crane lifts a steel beam of mass m = 450 kg to a height of h = 18 meters above the ground. Calculate the gravitational potential energy gained by the beam. Use g = 9.8 m/s².
View Step-by-Step Solution
- Identify the given values: Mass m = 450 kg, Height h = 18 m, and Gravity g = 9.8 m/s².
- Recall the gravitational potential energy formula: PE = m · g · h.
- Substitute the values into the equation: PE = 450 kg · 9.8 m/s² · 18 m.
- Multiply mass and gravity to get the force: F_g = 450 · 9.8 = 4,410 Newtons (weight of the beam).
- Calculate potential energy: PE = 4,410 N · 18 m = 79,380 Joules.
- Convert to kilojoules if needed: PE = 79.38 kJ.
A skateboarder of mass m = 60 kg starts from rest at the top of a frictionless U-shaped half-pipe at a height of h_initial = 5.0 meters. Using the conservation of mechanical energy, determine the skater's speed at the bottom of the ramp where height h = 0.
View Step-by-Step Solution
- Identify the values: Skater mass m = 60 kg, Initial height h_i = 5.0 m, Final height h_f = 0 m, Gravity g = 9.8 m/s².
- Recall the conservation of mechanical energy principle: Total mechanical energy is constant. E_initial = E_final.
- Write the energy equation: PE_initial + KE_initial = PE_final + KE_final.
- Simplify: Since the skater starts from rest, KE_initial = 0. Since h_final = 0, PE_final = 0. Therefore, PE_initial = KE_final.
- Substitute formulas: m · g · h_initial = 1/2 · m · v².
- Divide both sides by mass (m) to eliminate it: g · h_initial = 1/2 · v².
- Solve for velocity (v): v = √(2 · g · h_initial) = √(2 · 9.8 · 5.0) = √(98).
- Calculate final speed: v ≈ 9.90 m/s.
A pile driver hammer of mass m = 300 kg is raised to a height of h = 6.0 meters and dropped onto a structural metal pile. The pile is driven into the ground by a distance d_stop = 0.12 meters before the hammer is brought to rest. Solve for the average resistance force exerted by the soil. (Assume g = 9.8 m/s²)
View Step-by-Step Solution
- Identify given parameters: Hammer mass m = 300 kg, Drop height h = 6.0 m, stopping distance (pile penetration depth) d = 0.12 m, Gravity g = 9.8 m/s².
- Calculate the initial potential energy before release: PE = m · g · h = 300 · 9.8 · 6.0 = 17,640 Joules.
- Apply the Work-Energy Theorem: The work done by the resistive soil force (W_soil = -F_avg · d) must dissipate the total mechanical energy of the falling hammer.
- Set up the energy balance: Potential Energy gained equals Work done to stop: m · g · h = F_avg · d.
- Solve for average resistive force (F_avg): F_avg = (m · g · h) / d.
- Substitute values: F_avg = 17,640 J / 0.12 m = 147,000 Newtons.
- Convert to kilonewtons: F_avg = 147 kN.
Conceptual Practice
A book of mass 1.5 kg is moved from a shelf 2.0 meters high to a shelf 0.5 meters high. What is the change in its gravitational potential energy relative to the floor?
Show Explanation
The initial potential energy is PE_initial = mgh_initial = 1.5 · 9.8 · 2.0 = 29.4 J. The final potential energy is PE_final = mgh_final = 1.5 · 9.8 · 0.5 = 7.35 J. The change in potential energy is ΔPE = PE_final - PE_initial = 7.35 - 29.4 = -22.05 J. The book has lost 22.05 Joules of potential energy.
Explain why gravitational potential energy can be negative while kinetic energy is always positive.
Show Explanation
Kinetic energy depends on mass (always positive) and velocity squared (always positive), so KE >= 0. Gravitational potential energy depends on height (h) relative to a reference datum. If an object is positioned below the chosen reference level, its height h is negative, which yields a negative potential energy (PE = mgh < 0).
How does potential energy change if a satellite in a circular orbit doubles its distance from Earth's center? Explain why PE = mgh is not used here.
Show Explanation
For space scales, gravity is not constant. We must use Newton's general gravitational potential energy formula: U = -G*M*m/r. As distance r doubles, potential energy U increases (becomes less negative, approaching zero from below). The formula PE = mgh is only valid near Earth's surface where g is assumed constant.
A pile driver drops a 400 kg hammer from a height of 5 meters. If the soil stiffness doubles (meaning the pile penetrates half the distance compared to before), what happens to the average resistance force exerted by the soil?
Show Explanation
According to the relation F_avg · d_stop = mgh, if mass (m) and height (h) remain constant, the energy to be dissipated (mgh) is constant. If soil stiffness doubles and stopping distance (d_stop) is halved, the average resistive force F_avg must double to satisfy the equation (since F_avg = mgh / d_stop).
Frequently Asked Questions
What is potential energy?
Potential energy is the stored energy an object possesses because of its position, state, or configuration relative to other objects. For gravitational potential energy, this refers to an object's elevation above a chosen reference level.
What is the formula for potential energy?
The formula for gravitational potential energy is PE = mgh, where m is mass in kilograms (kg), g is the acceleration due to gravity (approximately 9.8 m/s2 on Earth), and h is the height in meters (m) above the reference level.
What does each variable in the PE = mgh formula mean?
In the equation PE = mgh: m is the mass of the object (kg), g is the local gravitational acceleration (9.8 m/s2 on Earth), and h is the vertical height or displacement of the object from a chosen datum (m).
What is the SI unit of potential energy?
The SI unit of potential energy is the Joule (J). One Joule is equivalent to 1 kg·m2/s2, which is also represented as a Newton-meter (1 N·m).
Why is potential energy considered relative?
Potential energy is relative because height (h) must be measured from an arbitrarily chosen reference level (called the datum or zero level), such as the ground, a tabletop, or sea level. Changing the reference level shifts the absolute value of PE, but the *change* in potential energy (Δ PE = mgΔ h) remains the same.
How does mass affect potential energy?
Potential energy scales linearly with mass (PE ∝ m). If you double the mass of an object while holding height and gravity constant, its potential energy doubles. If mass is tripled, PE triples.
How does height affect potential energy?
Potential energy scales linearly with height (PE ∝ h). Doubling the elevation of an object doubles its potential energy. Raising it three times higher triples its PE.
How does the value of gravity affect potential energy?
Potential energy is directly proportional to gravitational acceleration (PE ∝ g). An object at a height of 2 meters on Jupiter (g ≈ 24.79 m/s2) has much more potential energy than the same object at the same height on the Moon (g ≈ 1.62 m/s2).
What is the conservation of mechanical energy?
The conservation of mechanical energy states that in the absence of non-conservative forces (like friction or air resistance), the sum of potential and kinetic energy in a closed system remains constant: Etotal = PE + KE = constant.
What is a reference level or datum in physics?
A reference level or datum is the baseline position where the height h is defined as zero, making the potential energy at that point zero (PE = 0). It is chosen for convenience to simplify calculations for a specific problem.
How does potential energy relate to work?
The change in potential energy is equal to the negative of the work done by the conservative gravitational force: Δ PE = -Wg. Conversely, the work done by an external force to lift an object at constant speed is equal to the increase in its potential energy: Wext = Δ PE = mgh.
Can potential energy be negative?
Yes, potential energy can be negative if the object is located below the chosen reference level. For example, if the table surface is h = 0, an object resting on the floor below the table has negative potential energy relative to that table.