Потенційна енергія – проблеми та рішення

Потенційна енергія – проблеми та рішення

Гравітаційна потенційна енергія

1. Для підняття об'єкта масою 50 кг на висоту h використовується енергія 4900 джоулів. Яка висота h? Прискорення за рахунок сили тяжіння (g) = 9.8 мс^-2.

Відомо:

Зміна потенційна енергія (ΔPE) = 4900 Джоулів

Маса об'єкта (м) = 50 кг

Прискорення вільного падіння (g) = 9.8 м/с²

Бажано: height (Δh)

Рішення:

ΔPE = мг Δгод

4900 = (50)(9.8) Δh

4900 = 490 Δг

Δh = 10 meters

Spring potential energy

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?

Відомо:

Сила (F) = 2 Ньютонів

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

Розшукується: The spring potential energy

Рішення:

Spring’s constant :

The spring potential energy :

∆PE = 1/2 k x²

∆PE = 1/2 (200 N/m)(0.08 m)²

∆PE = (100 N/m)(0.0064 m²)

∆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.

Відомо:

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

Розшукується: Work done on the spring

Рішення:

Spring’s constant :

k = F / ∆x = 20 Н / 0.04 м = 500 Н/м

k = F / ∆x = 30 Н / 0.06 м = 500 Н/м

k = F / ∆x = 40 Н / 0.08 м = 500 Н/м

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 ∆L² = 1/2 (500 Н/м)(0.1 м)² = (250 Н/м)(0.01 м²) = 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.

Відомо:

F = 40 Ньютонів

Δx = 0.08 m

Розшукується: spring potential energy

Рішення:

Spring’s constant :

k = F / Δx

The spring potential energy :

PE = ½ k Δx² = ½ (F/Δx) Δx² = ½ F Δx

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

1. What is potential energy?
   • Відповідь: 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.
2. How does gravitational potential energy depend on height?
   • Відповідь: 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, $g$ – гравітаційне прискорення, а $h$ це висота.
3. Why is it said that a stretched or compressed spring has potential energy?
   • Відповідь: 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.
4. How does the potential energy of an object change if it’s raised to twice its original height in a gravitational field?
   • Відповідь: 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.
5. Is it possible for an object to have negative potential energy? Explain.
   • Відповідь: 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.
6. How is potential energy related to kinetic energy in a closed system?
   • Відповідь: 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.
7. What happens to the gravitational potential energy of an apple when it falls from a tree?
   • Відповідь: 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.
8. Why does a pendulum have the most potential energy at the peak of its swing?
   • Відповідь: 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.
9. Can an object have potential energy even if it’s not in a gravitational field?
   • Відповідь: 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.

10. How does the mass of an object influence its gravitational potential energy?

    • Відповідь: 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.