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One way to find a dot product is using

Using that here...

Problem 2Thursday, May 24, 2012

10:37 AM

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Thexinterval over which we are supposed to find work is actually

inconvenient. Because of this odd interval, it would probably be easiestjust to find the equation of the line and integrate. The equation for this

line is...

To find the work, we just integrate...

Problem 3Thursday, May 24, 201 2

10:37 AM

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We can actually do the same approach here, but the line is really

boring.

This gives a work of...

This is now a bit more interesting line...

Finding the work is the same approach as before...

We don't even have to use calculus for this, we can use geometry to

find the area under the curve.

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Because force increases uniformly with distance, this works just like a

regular spring. The relationship between force and displacement for a

spring is...

Solving for k...

To find the work, we just integrate the force over the displacement.

Problem 4Thursday, May 24, 2012

10:37 AM

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To find work, we integrate the force over the distance.

In terms of the formula, nothing has changed. Now we are just

evaluating the integral over a longer length.

Problem 5Thursday, May 24, 201 2

10:37 AM

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To find the percent difference between this answer and that of part

(a)...

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We know the work and we know the displacement. Because force is a

constant, the expression for work is...

We are told that the force applied in part (a) leads to a constant

velocity (no acceleration). This means that the workers force perfectly

balances with the frictional force.

If he exerts a larger force, the forces don't balance anymore and the

block will accelerate!

Also, if the force is smaller than that of part (a), the forces don't

Problem 6Thursday, May 24, 2012

10:37 AM

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balance anymore. In this case though, the acceleration will lead to the

block slowing down and stopping.

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Here, the gravitational potential energy of the child can be written as...

Each of these parts just has a different value of.

Problem 7Thursday, May 24, 2012

10:37 AM

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To find the force from a given potential, we can use the expression

These components can be put together to form the force vector.

Problem 8Thursday, May 24, 201 2

10:37 AM

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This is a standard conservation of energy problem. Because we are

asked to find the height from where it was released, we can set the zero

of gravitational potential energy at the release point. Then we will have

complete conversion of spring energy at the beginning to gravitational

potential energy at the end.

Problem 9Thursday, May 24, 2012

10:37 AM

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In this problem, the potential energy ofm1 will be converted to

potential energy m2 and kinetic energy ofm1 and m2.

Now we have a new system where we are only looking at m2. Because

we would like to have the maximum height as measured from the

Problem 10Thursday, May 24, 2012

10:37 AM

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table, let's use that as our reference point.

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One thing that we know from kinematics is that the x-component of

velocity never changes. When the y-component of velocity goes to

zero, we will still have a minimum kinetic energy.

The total kinetic energy at the start is

This means the maximum potential energy is...

Problem 11Thursday, May 24, 2012

10:37 AM

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If we fire straight up, we can use this same expression, but we replace

the .

In both of these systems, the mechanical energy is conserved. That

means the initial kinetic energy is the mechanical energy at every point

in time later (or at least until it hits the ground). This includes the

highest point.

This is true regardless of angle so it will be true for both of these

cannonballs.

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Initially in this problem, we have kinetic energy. Some of this energy isconverted to potential and some is lost to heat through friction.

Solving for the friction force, we get...

From the free-body diagram, we know that the force of friction will

be...

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The energy-efficient bulb has a cost consisting of bulb and energy over

its lifetime.

We will need a lot of conventional bulbs to handle this same time.

The cost of these bulbs are...

The total cost of operating the conventional bulb over the same time

period is the cost of the bulbs plus the cost of the energy.

In this problem, t/t2 is 13.3. Because this isn't an integer number, this

means we will need 14 bulbs.

The cost savings is...

Problem 13Thursday, May 24, 2012

10:38 AM

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If South Carolina wants to use conventional bulbs, they can go rightahead.