What is the Work-Energy Theorem and How Does It Show Conservation of Energy?

The Work-Energy Theorem is an important idea in physics that links work and energy. In simple words, this theorem says that the work done on an object changes its kinetic energy. For example, when you push a car, the energy from your push changes how fast the car goes. That’s work happening! Let’s take a closer look at how this theorem works and how it connects to the idea of conserving energy.

Understanding Work

First, we need to understand what "work" means in physics. Work happens when you apply a force to an object, making it move. You can calculate work with this formula:

$$W = F \cdot d \cdot \cos(\theta)$$

Here’s what each part means:

• ( W ) is the work done,
• ( F ) is how strong the force is,
• ( d ) is the distance the object moves in the direction of the force, and
• ( \theta ) is the angle between the force and the direction the object is moving.

For example, if you push a box across the floor, the work done on the box is the force of your push multiplied by how far the box moves. If you push with a force of 10 N for 5 m in the same direction, then:

$$W = 10 \, \text{N} \cdot 5 \, \text{m} = 50 \, \text{J}$$

Kinetic Energy

Kinetic energy is the energy of something that is moving. You can find it using this formula:

$$KE = \frac{1}{2} mv^2$$

Where:

• ( KE ) is kinetic energy,
• ( m ) is the mass of the object, and
• ( v ) is how fast it is moving.

Think about a skateboarder going down a hill. As they go faster, their kinetic energy gets bigger. If the skateboarder starts still and rolls down the hill, gravity does work on them and increases their kinetic energy.

The Work-Energy Theorem

The Work-Energy Theorem can be expressed like this:

$$W = \Delta KE = KE_f - KE_i$$

Here:

• ( \Delta KE ) is the change in kinetic energy,
• ( KE_f ) is the final kinetic energy, and
• ( KE_i ) is the initial kinetic energy.

So, if the skateboarder starts at a speed of 0 m/s and finishes at 5 m/s, we can figure out how much their kinetic energy changed and how it relates to the work done by gravity.

Conservation of Energy

Now, let’s connect this idea to energy conservation. The Work-Energy Theorem shows us a key point: energy is conserved in a closed system. The work done on an object causes a change in its kinetic energy. If there’s no energy lost to things like friction, the total energy (potential and kinetic) stays the same.

For example, think of a pendulum. At the highest point, it has lots of potential energy but no kinetic energy. As it swings down, some potential energy turns into kinetic energy. At the lowest point, it has the most kinetic energy and no potential energy. This change shows the conservation of energy:

$$PE_i + KE_i = PE_f + KE_f$$

Conclusion

The Work-Energy Theorem clearly shows how work done on an object affects its kinetic energy. By understanding this theorem, we can see how energy changes and is conserved in different physical situations. Whether it’s a skateboarder on a hill or a pendulum swinging back and forth, the ideas of work and energy help us understand how motion and force work in physics. So, the next time you see something speeding up or slowing down, think about the work done and how it connects to energy changes—it’s a fascinating part of physics in motion!