Work-Kinetic Energy principle

If net force works on an object, the object experiences acceleration (the object experiences displacement). When the object experiences acceleration, the speed of the object changes. In other words, the work done by the net force is related to the object’s initial and final speed.

The work done on an object by constant net force is:

W_{net} = ΣF s

Newton’s second law states that if there is a net force working on an object, the object experiences acceleration.

W_{net }= (m a) s

If the net force is constant, the acceleration experienced by the object is constant as well. Therefore, we can substitute a non-uniform linear motion equation for acceleration (a) and displacement (s).

Incorporate the non-uniform linear motion equation into the work equation:

Description: EK_{t }= final kinetic energy, EK_{o} = initial kinetic energy, m = mass, v_{t }= final speed, v_{o} = initial speed

This equation constitutes the work-kinetic energy theorem. The work-kinetic energy theorem informs us that net work or the work done by the net force on an object is equal to the change in the object’s kinetic energy. It also informs us that an object’s kinetic energy is equal to the net work required to accelerates the object from a stationary state to moving at a given speed, and vice versa.

Example question 5: Work-kinetic energy theorem

A car with a mass of 1000 kg moves from a stationary state. In an instant, the speed increases into 10 m/s. How much is the net work done by the car’s engines?

Solution :

Known: m = 1000 kg, v_{o} = initial speed = 0 m/s (at first, the car is at rest), v_{t} = final speed = 10 m/s

Wanted : The net work

W_{net} = 1⁄2 m (v_{t}^{2} – v_{o}^{2})

W_{net }= 1⁄2 (1000)(10^{2} – 0^{2}) = (500)(100 – 0) = (500)(100) = 50,000 Joule

**1. What is the Work-Kinetic Energy principle?** The Work-Kinetic Energy principle states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, $W_{net}=ΔKE$, where $ΔKE$ is the final kinetic energy minus the initial kinetic energy.

**2. How is work related to a change in velocity of an object?** Work done on an object can change its velocity. Specifically, the net work done on an object is equal to the change in its kinetic energy, and kinetic energy is related to the square of velocity. Thus, if positive net work is done on an object, its velocity (and therefore its kinetic energy) increases.

**3. What happens to the kinetic energy of an object when negative work is done on it?** When negative work is done on an object, its kinetic energy decreases. This could result in the object slowing down.

**4. How does the Work-Kinetic Energy principle relate to Newton’s second law?** Newton’s second law describes how a force affects an object’s motion, while the Work-Kinetic Energy principle relates the work done by that force to the change in kinetic energy. Both concepts are linked through the idea that forces can change an object’s state of motion and energy.

**5. Can the kinetic energy of an object be negative?** No, kinetic energy is always non-negative. It is given by the formula $KE= 1/2 mv$, where m is mass and v is velocity. Since both mass and the square of velocity are positive, kinetic energy is also positive or zero.

**6. How does the Work-Kinetic Energy principle apply to an object moving in a circular path?** For an object moving in a circular path at constant speed, its kinetic energy remains constant since its speed is unchanged. However, the direction of its velocity changes continuously. If only centripetal forces (like tension or gravitational force) act on the object, the work done by these forces is zero as they act perpendicular to the direction of motion. Thus, no net work is done, and there’s no change in kinetic energy.

**7. In the absence of external forces, how does the kinetic energy of an object change?** In the absence of external forces, no work is done on the object, which means its kinetic energy remains constant. This is a manifestation of the conservation of energy.

**8. What does it mean when the net work done on an object is zero?** When the net work done on an object is zero, it means the object’s kinetic energy hasn’t changed. The object could still be moving, but its speed remains constant.

**9. Can an object have kinetic energy even if no work has been done on it?** Yes, an object can have kinetic energy without any work being done on it in a given time frame. For instance, an object already in motion in outer space (far from gravitational bodies) will retain its kinetic energy and keep moving at a constant velocity, even if no external forces act on it.

**10. How does the work done by nonconservative forces affect the kinetic energy of an object?** The work done by nonconservative forces can increase or decrease the kinetic energy of an object, depending on the direction and magnitude of the force relative to the object’s motion. For instance, friction, a nonconservative force, usually does negative work, thereby decreasing an object’s kinetic energy.