Potential energy

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Potential energy

One more rule about conservative forces is that we can define a quantity called potential energy for them.

We define gravitational potential energy as .

We have found that the work due to gravity is where y1 is the initial height and y2 is the final height.

With this definition:

If gravity is the only force contributing to the work then and so by the work energy theorem we arrive at

Potential energy which can be written

We further define the total mechanical energy of the system as

Then the above equation is simply

Since points 1 and 2 are arbitrary what we are really saying is that E is constant no matter where we are.

So long as gravity is the only force acting, E is constant. This is an example of the conservation of mechanical energy.

The power of conservation of energy A block starts at rest at a height of 2.5 m and slides down a frictionless inclined plane. What is the block velocity when it reaches the bottom?

2.5 m

Initial energy: Final energy:

Conservation of energy:

If energy is conserved you don’t need to know how a particle gets from one place to another. If you can determine the energy at one place then you know it at any other place.

Clicker question 1 Set frequency to BA

A marble rolls down a frictionless track, and reaches speed v at the bottom.

If you want it to reach a speed of 4v at the bottom, you need the start of the new track to be A: twice as high B: 4 times as high C: half as high

D: 16 times as high E: (need more information)

as the original track height.

Clicker question 1 Set frequency to BA

A marble rolls down a frictionless track, and reaches speed v at the bottom.

If you want it to reach a speed of 4v at the bottom, you need the start of the new track to be A: twice as high B: 4 times as high C: half as high

D: 16 times as high E: (need more information)

as the original track height.

PE initial = mgh, KE final = 1/2 m v2. To reach 4 v, you need 42=16 times the energy. So, you need 16 times the height.

A.  yes B.  no C.  impossible to tell without knowing the mass

Clicker question 2 Set frequency to BA

A hockey puck slides without friction along a frozen lake toward an ice ramp and plateau as shown. The speed of the puck is 4 m/s and the height of the plateau is 1 m. Will the puck make it all the way up the ramp?

v = 4 m/s h = 1 m

A.  yes B.  no C.  impossible to tell without knowing the mass

Clicker question 2 Set frequency to BA

A hockey puck slides without friction along a frozen lake toward an ice ramp and plateau as shown. The speed of the puck is 4 m/s and the height of the plateau is 1 m. Will the puck make it all the way up the ramp?

The minimum speed needed to make it to the top is when it arrives at the top at rest (no kinetic energy).

v = 4 m/s h = 1 m

Conservation of energy:

Initial speed must be or greater to make it to the top. But

Clicker question 3 Set frequency to BA

A pendulum is launched in two different ways. During both launches, the bob has an initial speed of 3.0 m/s. On launch 1, the speed is up (along the trajectory). On launch 2, the speed is down (along the trajectory).

Which launch will cause the pendulum to swing the largest angle from the equilibrium position on the left side? A: Launch 1 B: Launch 2 C: Both launches give the same max displacement.

Clicker question 3 Set frequency to BA

A pendulum is launched in two different ways. During both launches, the bob has an initial speed of 3.0 m/s. On launch 1, the speed is up (along the trajectory). On launch 2, the speed is down (along the trajectory).

Which launch will cause the pendulum to swing the largest angle from the equilibrium position on the left side? A: Launch 1 B: Launch 2 C: Both launches give the same max displacement.

Conservation of energy says KE(init) is what matters in determining the height. Squaring the v means sign doesn't matter. If you swing it with v1, it'll go up and come back, and when it reaches its starting point again it'll have v2=-v1. Either way, it makes it just as far on the left side. (Barring frictional losses)

Clicker question 4 Set frequency to BA

A mass m is at the end of light (massless) rod of length R, the other end of which has a frictionless pivot so the rod can swing in a vertical plane.

The rod is initially horizontal and the mass is pushed down with an initial speed vo.

What initial kinetic energy is required for the mass to pivot 270o (to the vertical, or "12 o'clock" position?)

€

12mvo

2= A: mgR B: mg*(2R) C: mg*(3R) D: 0 E: None of these

Clicker question 4 Set frequency to BA

A mass m is at the end of light (massless) rod of length R, the other end of which has a frictionless pivot so the rod can swing in a vertical plane.

The rod is initially horizontal and the mass is pushed down with an initial speed vo.

What initial kinetic energy is required for the mass to pivot 270o (to the vertical, or "12 o'clock" position?)

€

12mvo

2= A: mgR B: mg*(2R) C: mg*(3R) D: 0 E: None of these

KE(i)+PE(i) = KE(f)+PE(f) At the top, KE(f) = 0, if it "just makes it". Let's set the center of the circle to PE(i)=0. So KE(i)+0 = 0 + PE(f) = 0 + mg*R. That's all you need.

More about gravitational potential energy If gravity is the only force doing work, conservation of energy means

Note that the y=0 point is arbitrary (but must be the same reference point for all points). This has some consequences:

1. Gravitational potential energy can be positive or negative.

2. Only differences in gravitational potential energy are physically meaningful.

Clicker question 5 Set frequency to BA A small mass, starting at rest, slides without friction down a loop-de-loop as shown. The maximum height of the loop is the same as the initial height of the mass.

Will the ball make it to the top of the loop? A: Yes, the ball makes it to the top of the loop. B: No, the ball will not make it to the top. C: Not enough info to say, or don't know.

Clicker question 5 Set frequency to BA A small mass, starting at rest, slides without friction down a loop-de-loop as shown. The maximum height of the loop is the same as the initial height of the mass.

Will the ball make it to the top of the loop? A: Yes, the ball makes it to the top of the loop. B: No, the ball will not make it to the top. C: Not enough info to say, or don't know.

This is subtle! It has enough energy to make it to the top, but as it climbs the loop, its velocity will slow, and it will come to a point when it falls off the loop. (At the peak of its trajectory, will it be at its original height?) No! At the top of the trajectory, it still has an "x-component" of velocity, i.e. some KE, which takes away from PE. Basically, its energy is never fully converted into pure PE - some of it will still be KE.

What about forces other than gravity? The spring force is also a conservative force

The force applied to the spring is while the force by the spring is Note that the x=0 point is not arbitrary; the x=0 point is when the spring is in the relaxed position

relaxed spring – no force applied

extended spring – displacement and applied force to the right

Work by a spring The work done by a spring is

This is true for positive or negative x1 and x2.

Spring is extended to position x2 so force by spring is opposite the motion and so work is negative:

A spring starts at x = 0

Then the spring is moved from x2 to –x2 and the positive work from x2 to 0 is canceled by the negative work from 0 to x2:

Elastic potential energy The potential energy from a spring is Remember x is the distance from the relaxed position

The combined conservation of mechanical energy equation is valid when only gravitational and elastic forces do work:

or simply

This potential energy is always ≥0 and is physically meaningful by itself (unlike gravitational potential energy)

Use conservation of energy to find the kinetic energy at impact

Using conservation of energy A spring loaded gun with spring constant of 5000 N/m is used to fire a 0.01 kg BB off a 100 m cliff. The spring is compressed 0.1 m before launch. Assuming no air resistance, what is the speed of the BB just before it hits the ground?

Initial energy: We choose to set y=0 at the base of the cliff

Final energy:

By conservation of energy:

Using conservation of energy A spring loaded gun with spring constant of 5000 N/m is used to fire a 0.01 kg BB off a 100 m cliff. The spring is compressed 0.1 m before launch. Assuming no air resistance, what is the speed of the BB just before it hits the ground?

What if we pick the y=0 point at the top of the cliff?

Initial energy:

Final energy:

Energy conservation:

Clicker question 6 Set frequency to BA

A spring-loaded dart gun shoots a dart straight up into the air, and the dart reaches a height of 24 m. The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the dart go this time, neglecting friction and assuming an ideal spring?

A.  3 m B.  6 m C.  12 m D.  24 m E.  48 m

Clicker question 6 Set frequency to BA

A spring-loaded dart gun shoots a dart straight up into the air, and the dart reaches a height of 24 m. The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the dart go this time, neglecting friction and assuming an ideal spring?

A.  3 m B.  6 m C.  12 m D.  24 m E.  48 m

Before the dart is launched and when it reaches the maximum height the kinetic energy is 0

Initial energy: Final energy:

By conservation of energy, E1 = E2

If x is reduced by a factor of 2, h will be reduced a factor of 4