Potential energy – problems and solutions

**Gravitational potential energy**

1. Energy 4900 Joule used to raise an object with mass of 50 kg to a height of h. What is the height of h? Acceleration due to gravity (g) = 9.8 ms^{-2}.

__Known :__

Change of potential energy (ΔPE) = 4900 Joule

Mass of object (m) = 50 kg

Acceleration due to gravity (g) = 9.8 m/s^{2}

__Wanted:__ height (Δh)

__Solution :__

ΔPE = m g Δh

4900 = (50)(9.8) Δh

4900 = 490 Δh

Δh = 10 meters

2. The graph below shows the relation between force (F) and x (the change in length) of a spring. If the change in length of a spring is 8 cm, what is the spring potential energy?

__Known :__

Force (F) = 2 Newton

The change in length 1 (x) = 1 cm = 1/100 m = 0.01 m

The change in length 2 = 8 cm = 8/100 m = 0.08 m

__Wanted :__ The spring potential energy

__Solution :__

Spring’s constant :

The spring potential energy :

∆PE = 1/2 k x^{2}

∆PE = 1/2 (200 N/m)(0.08 m)^{2}

∆PE = (100 N/m)(0.0064 m^{2})

∆PE = 0.64 Nm

3. Based on table below, F = weight of object, ∆L = the change in length of spring. What is the work done on the spring so the change in length of spring is 10 cm.

__Known :__

The change in length of spring (∆L) = 10 cm = 0.1 m

__Wanted :__ Work done on the spring

__Solution :__

Spring’s constant :

k = F / ∆x = 20 N / 0.04 m = 500 N/m

k = F / ∆x = 30 N / 0.06 m = 500 N/m

k = F / ∆x = 40 N / 0.08 m = 500 N/m

Spring’s constant is 500 N/m

The work done on the spring so the change in length of spring is 10 cm :

W = 1/2 k ∆L^{2} = 1/2 (500 N/m)(0.1 m)^{2} = (250 N/m)(0.01 m^{2}) = 2.5 N m = 2.5 Joule.

4. Graph below shows relation between force (F) and the change in length (x). What is the spring potential energy based on graph below.

__Known :__

F = 40 Newton

Δx = 0.08 m

__Wanted :__ spring potential energy

__Solution :__

Spring’s constant :

k = F / Δx

The spring potential energy :

PE = ½ k Δx^{2 }= ½ (F/Δx) Δx^{2 }= ½ F Δx

PE = ½ (40)(0.08) = (20)(0.08) = 1.6 Joule

**What is potential energy?****Answer**: Potential energy is the energy an object has because of its position or state. It’s the energy that has the potential to do work but isn’t causing motion at the moment.

**How does gravitational potential energy depend on height?****Answer**: Gravitational potential energy is directly proportional to the height of an object above a reference point. The formula is PE$=mgh$, where PE is the potential energy, $m$ is mass, $g$ is the gravitational acceleration, and $h$ is the height.

**Why is it said that a stretched or compressed spring has potential energy?****Answer**: When a spring is stretched or compressed from its equilibrium position, it stores energy. This energy has the potential to do work when the spring returns to its equilibrium state. This stored energy is referred to as elastic potential energy.

**How does the potential energy of an object change if it’s raised to twice its original height in a gravitational field?****Answer**: If an object is raised to twice its original height, its gravitational potential energy doubles. This is because the potential energy is linearly dependent on height.

**Is it possible for an object to have negative potential energy? Explain.****Answer**: Yes, potential energy can be negative depending on the choice of reference point. For example, in gravitational potential energy problems, we often set the ground as having zero potential energy. If an object were below this reference point (like in a well), its potential energy would be negative relative to the chosen reference.

**How is potential energy related to kinetic energy in a closed system?****Answer**: In a closed system with no external forces, the total energy (kinetic + potential) remains constant. This principle is based on the conservation of energy. As potential energy increases, kinetic energy decreases, and vice versa.

**What happens to the gravitational potential energy of an apple when it falls from a tree?****Answer**: As the apple falls, its gravitational potential energy decreases because it’s getting closer to the Earth. This lost potential energy is converted into kinetic energy, making the apple move faster as it falls.

**Why does a pendulum have the most potential energy at the peak of its swing?****Answer**: At the peak of its swing, the pendulum is at its highest point relative to its resting position. This means it has the maximum height and, therefore, the maximum gravitational potential energy. As it swings down, this energy gets converted to kinetic energy.

**Can an object have potential energy even if it’s not in a gravitational field?****Answer**: Yes. Potential energy is not exclusive to gravitational fields. For instance, a compressed spring on a space station (where gravitational effects are minimal) still has elastic potential energy. Similarly, charges in an electric field have electric potential energy.

**How does the mass of an object influence its gravitational potential energy?****Answer**: Gravitational potential energy is directly proportional to the mass of an object. If you double the mass of an object, its gravitational potential energy (at the same height) will also double.

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