Kumumanaʻo ikehu hana-mekanika

Kumumanaʻo ikehu hana-mekanika

ʻŌlelo ke kumumanaʻo ikehu hana-kinetic ua like ka hana upena a i ʻole ka hana i hana ʻia e ka ikaika upena me ka loli o ka ikehu kinetic.

W_upena = I_t – IĀ_o = 1⁄2 m(v_t² - v_o²)

W_upena = Aia ʻelua ʻano o nā ikaika, ʻo ia hoʻi ka ikaika conservative, a me ka ikaika non-conservative. No laila, hiki ke manaʻo ʻia ka hana ʻupena he hana i hana ʻia e kahi ikaika conservative a me ka hana i hana ʻia e kahi ikaika non-conservative.

W_c + W_nc = ΔKE

The work done by a conservative force is equal to the negative change in potential energy:

W_c= -ΔPE

– ΔPE + W_nc = ΔKE

W_nc = ΔPE + ΔKE

Wnc = ΔME

The equation above states that the work done by a non-conservative force on an object is equal to the change in the mechanical energy of the object. Mechanical energy = potential energy + kinetic energy. Potential energy can take the form of gravitational potential energy or elastic potential energy.

Example question: The work-mechanical energy theorem

A 2 kg box initially moves at a speed of 10 m/s. Shortly after, the box stops. The coefficient of kinetic friction between the box and the floor is 0.2. The gravitational acceleration is 10 m/s². How much is the box’s displacement?

Kōkua:

Identified: m = 2 kg, v_o = 10 m/s, vt = 0, k = 0.2, w = m g = (1 kg)(10 m/s²) = 10 kg m/s² = 10 Newton,

Asked: the amount of the box’s displacement (s)

The work-mechanical energy theorem:

W_nc = ΔME

W_nc = ΔPE + ΔKE

The height (h) remains constant or there is no change in the height, so there is no change in the gravitational potential energy.

Wnc = ΔKE

The work done by the kinetic frictional force is:

W_nc = – f_k s = μ_k N -s = μ_k w -s = μk m g -s

W_nc = – (0.2)(2)(10)(s) = – (4)(s)

The kinetic frictional force does negative work (the kinetic frictional force is in opposite direction from the object’s displacement)

Change in the kinetic energy:

ΔKE = 1⁄2 m (v_t² - v_o²) = 1⁄2 (2)(0² - 10²) = (0 – 100) = – 100

Object’s displacement:

W_nc = ΔKE

– (4)(s) = – 100

s = – 100 / – 4 = 25 meters

20 conceptual questions and answers about the work-mechanical energy principle:

1. Nīnau: What is the work-mechanical energy principle? pane mai: The work-mechanical energy principle states that the work done on an object is equal to the change in its mechanical energy.

2. Nīnau: How is work defined in physics? pane mai: Work is defined as the product of the force applied to an object and the distance it moves in the direction of the force

3. Nīnau: What constitutes mechanical energy? pane mai: Mechanical energy consists of an object’s kinetic energy and potential energy.

4. Nīnau: In the absence of non-conservative forces, what can be said about the total mechanical energy of a closed system? pane mai: ʻO ka ikehu mechanical holoʻokoʻa e mau ana.

5. Nīnau: Give an example of a conservative force. pane mai: Gravitational force is an example of a conservative force.

6. Nīnau: How is potential energy different from kinetic energy? pane mai: Potential energy is the energy due to position, like height above the ground, while kinetic energy is due to motion.

7. Nīnau: How does the work done by a conservative force relate to potential energy? pane mai: Ua like ka hana i hana ʻia e kahi ikaika conservative me ka loli maikaʻi ʻole o ka ikehu hiki.

9. Nīnau: How does the work-mechanical energy principle explain the conversion of potential energy to kinetic energy in a free-falling object? pane mai: As an object falls, its potential energy decreases and its kinetic energy increases by an equivalent amount, keeping the total mechanical energy constant.

10. Nīnau: What happens to the mechanical energy of a system when non-conservative forces, like friction, act on it? pane mai: The mechanical energy decreases because non-conservative forces dissipate energy, often as heat.

11. Nīnau: Can the mechanical energy of a system increase? pane mai: Yes, when external work is done on the system.

12. Nīnau: How is the concept of work related to energy? pane mai: Work is the means by which energy is transferred or transformed in a system.

13. Nīnau: If an object is moving at a constant velocity, what can be said about the net work done on it? pane mai: The net work done on it is zero because no acceleration is happening.

15. Nīnau: Why does a compressed spring have potential energy? pane mai: When a spring is compressed, work is done on it, which is stored as potential energy. This energy can be released when the spring is let go.

16. Nīnau: What is the work-energy theorem? pane mai: The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.

17. Nīnau: How is power related to work? pane mai: Power is the rate at which work is done or energy is transferred. It’s calculated as $P=W/t$, kahi $W$ is work and $t$ is time.

18. Nīnau: Can work have a negative value? pane mai: Yes, work is negative when the force and the direction of motion are opposite.

19. Nīnau: If the angle between the force and the direction of motion is 90 degrees, what is the work done? pane mai: The work done is zero because the force has no component in the direction of motion.

20. Nīnau: How does a simple machine, like a lever, make work easier? pane mai: A lever doesn’t change the total work done but changes the way the force is applied, making the task more manageable or efficient.

The concept of work and the mechanical energy principle are fundamental in physics and provide a foundational understanding of how energy is transferred and transformed in various systems.