Sheria ya Uhifadhi wa Nishati ya Mitambo ni nini

Swali

Nishati ya mitambo ni jumla ya uwezo na nishati ya kinetic katika mfumo. Kanuni ya uhifadhi wa nishati ya mitambo inasema kwamba jumla ya nishati ya mitambo katika mfumo (i.e., jumla ya uwezo pamoja na nguvu za kinetic) remains constant as long as the only forces acting are conservative forces. We could use a circular definition and say that a conservative force as a force which doesn’t change the total mechanical energy, which is true, but might shed much light on what it means.

The conservation of mechanical energy

A good way to think of conservative forces is to consider what happens on a round trip. If the kinetic energy is the same after a round trip, the force is a conservative force, or at least is acting as a conservative force. Consider gravity; you throw a ball straight up, and it leaves your hand with a certain amount of kinetic energy. At the top of its path, it has no kinetic energy, but it has a potential energy equal to the kinetic energy it had when it left your hand. When you catch it again it will have the same kinetic energy as it had when it left your hand. All along the path, the sum of the kinetic and potential energy is a constant, and the kinetic energy at the end, when the ball is back at its starting point, is the same as the kinetic energy at the start, so gravity is a conservative force.

Kinetic friction, Kwa upande mwingine, is a non-conservative force, Katika hali yoyote halisi. Katika hali yoyote halisi; a non-conservative force changes the mechanical energy, so a force that increases the total mechanical energy, Katika hali yoyote halisi, Katika hali yoyote halisi.

An example

Consider a person on a sled sliding down a 100 m long hill on a 30° incline. The mass is 20 Kwa Nini Bandari Mihuri Huogelea Juu Juu Chini, and the person has a velocity of 2 m/s down the hill when they’re at the top. How fast is the person traveling at the bottom of the hill? All we have to worry about is the kinetic energy and the gravitational potential energy; when we add these up at the top and bottom they should be the same, because mechanical energy is being conserved.

At the top: PE = mgh = (20) (9.8) (100sin30°) = 9800 J
KE = 1/2 Unda na usanidi kikoa chako na vikoa vidogo vyote2 = 1/2 (20) (2)2 = 40 J
Total mechanical energy at the top = 9800 + 40 = 9840 J

At the bottom: PE = 0 KE = 1/2 Unda na usanidi kikoa chako na vikoa vidogo vyote2
Total mechanical energy at the bottom = 1/2 Unda na usanidi kikoa chako na vikoa vidogo vyote2

If we conserve mechanical energy, then the mechanical energy at the top must equal what we have at the bottom. This gives:

1/2 Unda na usanidi kikoa chako na vikoa vidogo vyote2 = 9840, so v = 31.3 m/s.

Modifying the example

Now let’s worry about friction in this problem. Let’s say, because of friction, the velocity at the bottom of the hill is 10 m/s. How much work is done by friction, and what is the coefficient of friction?

The sled has less mechanical energy at the bottom of the slope than at the top because some energy is lost to friction (the energy is transformed into heat, kwa maneno mengine). Sasa, the energy at the top plus the work done by friction equals the energy at the bottom.

Energy at the top = 9840 J

Energy at the bottom = 1/2 Unda na usanidi kikoa chako na vikoa vidogo vyote2 = 1000 J

Kwa hiyo, 9840 + work done by friction = 1000, so friction has done -8840 J worth of work on the sled. The negative sign makes sense because the frictional force is directed opposite to the way the sled is moving.

How large is the frictional force? The work in this case is the negative of the force multiplied by the distance traveled down the slope, ambayo ni 100 m. The frictional force must be 88.4 N.

To calculate the coefficient of friction, a free-body diagram is required.

In the y-direction, there is no acceleration, KOZI:

The coefficient of kinetic friction is the frictional force divided by the normal force, so it’s equal to 88.4 / 169.7 = 0.52.