Conservation of Energy

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. The point where the block is about to leave the track is when N = 0. So at point A, or

Solution of Loop de Loop What is the minimum height h for which the block will reach point A on the loop without leaving the track?

At point A Newton’s 2nd law gives:

Applying conservation of energy

Another Example of KE Conservation What kinetic energy must a baseball on the end of a 4 m string have at the bottom in order that it makes it all the way around without the rope becoming slack?

Conservation of energy: E1 = E2. At the bottom the ball has kinetic energy and at the top, kinetic and potential energy.

At top: Limit where rope becomes slack is T=0.

So, at the top, Applying conservation of energy:

From Newton’s 2nd law, the maximum acceleration will occur wherever the net force is a maximum.

Clicker question 1 Set frequency to BA

A.  0 B.  M C.  E D.  Impossible to tell

A mass is oscillating back and forth on a spring as shown (in the picture the spring is fully compressed). At which position is the magnitude of the acceleration of the mass a maximum? Position 0 is the equilibrium (unstretched) position of the mass.

0 M E

For a spring so the maximum force occurs at the maximum value for x (relative to the equilibrium position)

Conservation of energy so far

This is valid when only gravitational and elastic forces do work

Work done by another force adds another term

If Wother>0 then energy will be added

If Wother<0 then energy will be lost

More conservation of energy

The main example for Wother is the work done by friction

New conservation of energy:

Wother is usually work done by friction which is negative so the final energy E2 will be less than the initial energy E1 in this case

Where does this energy go? Mostly it gets converted to heat (think about rubbing your hands together) Can also be converted to sound (think about clapping your hands together)

Can be converted to other forms of energy as well

More conservation of energy New conservation of energy:

Book introduces internal energy which represents the heating up of the object and environment and sets

If we could measure the temperature changes we could approximately determine ΔUint. Usually we need to calculate Wother instead so this definition doesn’t help much.

Nevertheless, with this definition we can write another conservation of energy equation

A.  h B.  more than h C.  less than h D.  impossible to tell without the mass

Clicker question 2 Set frequency to BA

A mass slides down a frictionless ramp of height h and hits a carpet with kinetic friction coefficient µk = 1.0. Its initial speed is zero. How far does the mass slide along the carpet?

h

Conservation of energy: Wother is work by friction

Initial energy: Final energy:

Getting force from potential To derive the gravitational and elastic potential energy we started from the work done and used

Let us consider this equation in one dimension Note the force and potential energy may depend on position So in 1-D becomes

and therefore (this looks like a slope)

Force–potential energy relationship Force acts to reduce potential energy

Check if it works for spring potential:

In 3-D the relationship is

Partial derivatives are simpler than total derivatives because you just take the derivative assuming the other variables are constant.

Check if it works for gravity:

Graphs of potential energy Elastic potential energy is given by a parabola equation:

The force associated with the potential energy is in the direction which reduces the potential energy.

At the minimum the force is zero and object is in a stable equilibrium position

Example is a ball in the bottom of a bowl

Can also have more complicated potential energy curves

Graphs of potential energy

Any maxima are places where the forces are zero but is, in fact, an unstable equilibrium

Can get potential energy for any position using graph. From E=K+U can determine K or E given the other.

Example is a ball on top of an inverted bowl

Clicker question 3 Set frequency to BA

A.  8 kJ B.  23 kJ C.  35 kJ D.  45 kJ E.  None of these

A cart rolls without friction along a track. The graph of U vs. position is shown. The total mechanical energy is 45kJ.

0 40 80 120 160 x(m)

50 40 30 20 10 0

Etot

U(kJ)

To within 3 kJ, what is the maximum kinetic energy over the stretch of track shown?

Since E=K+U, the maximum K is when U is a minimum which is for 120<x<140 where U=10 kJ so

Example – Diatomic Molecule

Diatomic molecules typically have a potential energy as a function of separation that looks something like this:

The potential energy is large and positive when x is small - so, if you squeeze it, the energy is large (think of a spring, with lots of stored energy when squeezed.) If x is large (atoms far apart), energy is basically zero. U is negative in the middle, that means the molecule is happy there! (Everything likes to go to the lowest possible energy, like balls rolling down to the basement.)

F= -dU/dx. For small x, dU/dx (slope) is -, the force is + (repulsive). (Shows the molecule acts like a spring.) For large x, dU/dx is +, the force is -, atoms are attracted together. If x=x0, the slope is zero: no force. This is the position where the molecule likes to be.

Final Energy Comments Energy is always conserved. If there’s friction, mechanical energy is “lost”, but surfaces heat up. Mechanical energy has been transformed into heat energy. Mechanical energy, K+U, is not conserved if there’s friction, but total energy, K+U+thermal+chemical+nuclear+rest mass energy+... is conserved. You have to keep track of all forms of energy, and then energy is exactly conserved.

That last term, “rest mass energy”, is given by E=mc2. Einstein discovered it, it’s another form of energy which is very real, and needs to be included if you want to use conservation of energy accurately. Unlike Newton’s laws (which were modified by relativity and quantum physics), conservation of total energy is still believed to be absolute and fundamental. The universe has a certain amount of energy which will never change. The form of energy can change, from kinetic to potential to heat, back and forth, but the total remains fixed.

Introduction to momentum We define a quantity called momentum as

This also gives us a new way to write Newton’s 2nd law:

Can also write out components:

and therefore

Introduction to momentum

This gives us the impulse–momentum theorem

For a constant force, the impulse is defined as

If we consider a constant force over a finite time interval then Newton’s 2nd law can be written

The change in momentum of a particle over a time interval is given by the impulse of the net force over the time interval.

Clicker question 4 Set frequency to BA

A. ¼ the momentum of the heavy cart B.  ½ the momentum of the heavy cart C.  equal to the momentum of the heavy cart D.  twice the momentum of the heavy cart E.  four times the momentum of the heavy cart

Consider two carts, of masses m and 2m, at rest on an air track. You push each cart for 3 s, exerting equal force on each. At the end, the momentum of the light cart is…

By the impulse-momentum theorem, the change in momentum is given by the impulse which is . The same force is applied over the same time, so the same momentum change must occur

Getting impulse from force vs time graph Just like getting displacement from velocity vs time and work from force vs distance, can also get impulse from the area under the force vs time curve

One rectangle is a small force over a long time and the other rectangle is a large force over a small time. Both have the same area and so give the same impulse

F

t

Momentum and kinetic energy Momentum and kinetic energy are both functions of the mass and velocity of an object. How do they differ?

Kinetic energy is a scalar, there is no directionality Momentum is a vector, has magnitude and direction

Kinetic energy comes from force applied over distance Momentum comes from force applied over time