Mechanical Energy and Work – Let’s Explore How Work Changes Mechanical Energy!

　　vector display : velocity 　 external force 　　　Reset
　　　　　　　　　　　　　

Description :
    This page is for checking the relationship between mechanical energy and work.
    We will look at the case of moving an object attached to one end of a spring while gravity acts vertically downward. By moving the hand that supports the other end of the spring, we can make the object undergo various motions. You will observe that the values of "kinetic energy of the object K" (green graph), "spring potential energy Ue" (gray graph), and "gravitational potential energy Ug (with the reference set to the gray line)" (brown graph) change continuously. (Initially, ignore the graph for W(P).) "K" changes in proportion to the square of the object's speed. "Ue" changes in proportion to the square of the spring's stretch or compression (the natural length of the spring is defined as half the length when the object is at rest in equilibrium with gravity). "Ug" changes in proportion to the height from the reference line, becoming negative below the reference line (purple graph). Let's observe the relationship between the motion of the object and the changes in each of its energy forms.
    Next, stop moving the hand. The object continues to move, and the values of the individual energy components continue to change, but you can confirm that the total mechanical energy "E" (red framed graph on the far right) remains constant. For example, when the object moves downward, "Ug" decreases, but "Ue" and "K" increase by the same amount, so the sum remains unchanged. Even though the individual energies increase or decrease, the total mechanical energy does not change and is conserved. This is known as the "law of conservation of mechanical energy."
    Now, try moving the hand again. You will see that the mechanical energy changes. This occurs because the external force exerted by the hand on the spring does work, causing a change in the mechanical energy. You can also observe the "work done by external force W" (which corresponds to the rate of work P over time) on the graph. When positive work (red graph) is done, the mechanical energy increases by that amount, and when negative work (blue graph) is done, the mechanical energy decreases. This confirms that the law of conservation of mechanical energy holds when no external force does work.
    At the bottom, you can check the "vector display" checkbox to display the object's velocity vector (green) and the external force vector exerted by the hand (red). By moving the hand in the direction of the external force vector, you can do positive work, and by moving it in the opposite direction, you can do negative work. This allows you to control the increase or decrease of mechanical energy.

● By moving the hand, you can make the object move through the spring.
● The graph on the right allows you to check the work W (or power rate P) done by the external force exerted by the hand, the kinetic energy K, the spring potential energy Ue, the gravitational potential energy Ug, and the mechanical energy E. The graph for mechanical energy also shows the breakdown of each energy component in different colors.
● By checking the "vector display" checkbox, you can display the velocity vector of the object and the external force vector exerted by the hand.
● The "Reset" button allows you to return to the initial state.

Note: Forces like gravity and elastic force that can be associated with potential energy are called conservative forces. Conservative forces store work done by external forces as potential energy, and conversely, when the potential energy decreases, the force can do exactly that amount of work on the outside. On the other hand, when work is done by a conservative force on an object, the object's kinetic energy increases by the same amount. Therefore, the mechanical energy, which is the sum of kinetic energy and potential energy, does not change, as any increase in one energy is exactly canceled out by the decrease in the other.