### Work and energy

There is however a problem here—obviously one can imagine collisions in which the “total amount of motion”, as defined above, is definitely not the same. Work, Energy and Power: Problem Set. Problem 1: Renatta Gass is out with her friends. Misfortune occurs and Renatta and her friends find themselves getting a . A conservation of energy problem where all of the energy is not conserved.

If the angle is 0 degrees, then the work term will be positive. The above equation is expresses the quantitative relationship between work and energy. This equation will be the basis for the rest of this unit. It will form the basis of the conceptual aspect of our study of work and energy as well as the guiding force for our approach to solving mathematical problems.

A large slice of the world of motion can be understood through the use of this relationship between work and energy. Raising a Barbell Vertically To begin our investigation of the work-energy relationship, we will investigate situations involving work being done by external forces nonconservative forces. Consider a weightlifter who applies an upwards force say N to a barbell to displace it upwards a given distance say 0. The initial energy plus the work done by the external force equals the final energy.

The final amount of mechanical energy J is equal to the initial amount of mechanical energy J plus the work done by external forces J. Catching a Baseball Now consider a baseball catcher who applies a rightward force say N to a leftward moving baseball to bring it from a high speed to a rest position over a given distance say 0.

The final energy 5 J is equal to the initial energy J plus the work done by external forces J.

A Skidding Car Now consider a car that is skidding from a high speed to a lower speed. The force of friction between the tires and the road exerts a leftward force say N on the rightward moving car over a given distance say 30 m.

## Momentum, Work and Energy

The final energy 80 J is equal to the initial energy J plus the work done by external forces J. Pulling a Cart Up an Incline at Constant Speed As a final example, consider a cart being pulled up an inclined plane at constant speed by a student during a Physics lab. The work-energy principle There is a strong connection between work and energy, in a sense that when there is a net force doing work on an object, the object's kinetic energy will change by an amount equal to the work done: Note that the work in this equation is the work done by the net force, rather than the work done by an individual force.

Gravitational potential energy Let's say you're dropping a ball from a certain height, and you'd like to know how fast it's traveling the instant it hits the ground.

### Momentum, Work and Energy

You could apply the projectile motion equations, or you could think of the situation in terms of energy actually, one of the projectile motion equations is really an energy equation in disguise. If you drop an object it falls down, picking up speed along the way. This means there must be a net force on the object, doing work.

This force is the force of gravity, with a magnitude equal to mg, the weight of the object. The work done by the force of gravity is the force multiplied by the distance, so if the object drops a distance h, gravity does work on the object equal to the force multiplied by the height lost, which is: An object with potential energy has the potential to do work. In the case of gravitational potential energy, the object has the potential to do work because of where it is, at a certain height above the ground, or at least above something.

Spring potential energy Energy can also be stored in a stretched or compressed spring.

An ideal spring is one in which the amount the spring stretches or compresses is proportional to the applied force. This linear relationship between the force and the displacement is known as Hooke's law.

For a spring this can be written: The larger k is, the stiffer the spring is and the harder the spring is to stretch. Occasionally we shall also need W for the magnitude of W, but you will know from context which is which. The motion of the bags is slow, their accelerations are small compared to g, so the force required to accelerate them is small compared to their weight.

If you remember the scalar productyou'll know that i. The joule symbol J is the SI unit of work: The joule is not very big on a human scale: It's possible for a fit person to do a megajoule of work in an hour.

## Mechanics: Work, Energy and Power

The pile is now shorter so I must lift the second bag through a larger increase in height: Work is also proportional to the force, so lifting two bags requires twice the force and I do twice as much work as on the first bag.

Hmm, I've not planned this well and must lift the last two bags further: Look at the big displacement of the trolley and how easy it is. The trolley is supporting six bags, so the force it applies upwards has magnitude 1. But the force is in the j or y direction and the displacement in the i or x direction: So no work is done.

But not if we use pulleys which are discussed in more detail in the physics of blocks and pulleys. Here, a single rope goes from the support, down to my harness, round the pulley, back to the support, round another pulley and back to my hands. The pulleys turn easily, so the tension T in each section of the rope is the same. There are three sections pulling me upwards.

From Newton's second law, the total force acting on me equals my mass times my acceleration.

### Analysis of Situations Involving External Forces

Compared with g, my acceleration here is negligible. So the force I need supply with my arms is reduced. How is this possible? Like levers, blocks and pulleys don't save you work, but they can reduce or increase the force, which can make a task more convenient and comfortable. Kinetic energy and the work energy theorem The multimedia tutorial presents this theorem, but perhaps you'd like to see it again here. Let's apply a constant force F to a mass m as it moves, in one dimension, a distance x.

It might, for instance, be the magnetic force that we used in our section on Newton's laws. Once we relate this to time and velocity, we shall have to do the integration.

Remember that there is help with calculus. Let's use the equations from one-dimensional kinematics for which there is a multimedia tutorial. Let's suppose that a body starts from rest and that we apply a constant total force F for a certain time T in one example, and for twice that time 2T in a later example the black graphs at right. The distance travelled while the force is acting, i.

- View Audio Guided Solution for Problem:
- Momentum Conservation and Newton’s Laws
- Work, Energy and Power: Problem Set

So the constant force has been applied over four times the distance, and has done four times the work. So, even though the velocity has only doubled, we have done four times as much work blue graph at right.

That is an important consequence: This has important implications for road safety, as we see next. Stopping distances and the work energy theorem If I travel twice as fast on my bicycle, how much further does it take to stop? I include only the distance after I apply the brakes, not the time it takes me to react to danger and to apply the brakes.

So, to do four times as much negative work, the braking force assumed constant must be applied over four times the distance. Please remember this on the road.

Potential energy Suppose I slowly lift a mass in a gravitational field. In this clip from the multimedia tutorial, the rope, with a little assistance from me, is slowly lifting a container of water.