1. Quiz 2 Average 26 Distribution
Min 2, Max 50
00-10: 6
11-20: 30
21-30: 33
31-40: 25
51-50: 12
It you need to see your point status please see me after class or in my office.
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2. Test Review: Problem 1
5 points: The ratio of the maximum acceleration to the maximum velocity (amax/vmax) for an object in simple harmonic motion is:
a) k b) w c) d) 1/w
Answer:
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3. Test Review: Problem 2 Pull
5 points: A mass of 10.0kg is at rest on a horizontal surface, the coefficient of static friction between the mass and surface is What horizontal force must be applied to move the mass?
a) 32.3N b)98.0N c)1N d)297N
Answer: mg FN
10kg
Pull
Ffr y x
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4. Test Review: Problem 3
5 points: A wheel of radius 0.15m is rotating at an angular velocity of 2.2 rads/sec. What is the velocity of the edge of the wheel?
a) 1m/s b)0.068m/s c)14.7m/s d)0.33m/s
Answer:
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5. Test Review: Problem 4
5 points: On a planet with a radius 1.5 times greater than that of earth but with the same mass, what would be the acceleration of gravity at the surface?
a) 4.36 m/s2 b) 0.23 m/s2 c) 6.53 m/s2 d) 0.15 m/s2
Answer:
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6. Test Review: Problem 5
5 points: Suppose planet A and planet B are each orbited by satellites at the same radii r. However the velocity of the satellite orbiting planet A is four times that of the velocity of the satellite orbiting planet B. What is mA/mB , the ratio of the mass of planet A to planet B?
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7. Test Review: Problem 6
10 points: A space laboratory is rotating to create artificial gravity, a person standing in a room with the floor at a radius of 2150m feels entirely at home since the centripetal acceleration is equal to g. What is the velocity of the person and angular velocity of the laboratory?
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8. Test Review: Problem 7 y x FN v0=5m/s m (+5pnts) Ffr mg
15 points: A sled is traveling at 5.00 m/s along a horizontal stretch of snow. The coefficient of kinetic friction is mk= Draw the freebody diagram for the sled. What is the sled’s acceleration and how far does it travel in the horizontal direction before stopping? mg
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9. Unit 4: Conservation of Energy
Definition of Work (7-1, 7-2, 7-3)
Examples of Work, Definition of Energy and the link to Work (7-3, 7-4, 7-5)
Potential Energy (8-1, 8-2)
Problem Solving with the Conservation of Energy (8-3, 8-4)
Applications of Conservation of Energy (8-5, 8-6, 8-7, 8-8, 8-9).
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10. Generalized Potential Energy
For gravity the potential energy is
And the work is given by a line integral
This can be generalized for any conservative force
But this does not hold for non-conservative forces such as friction because the integral would depend on the path and not the endpoints.
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11. Elastic Potential Energy
When uncompressed or stretched the end of the spring at the right rests at x=0.
When decompressed the spring can increase the velocity of the ball and in other words, do work on the ball.
According to Hooke’s Law, if a person is to compress the spring he or she must press with a force FP=+kx.
According to the 3rd Law the spring pushes back with FS=-kx.
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12. Elastic Potential Energy
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13. Comments on Potential Energy
Potential energy stored by a system has the potential to do work at a later time
Gravity: A brick held aloft - U=mgh
Elastic: A compressed and locked spring- U = (1/2)kx2
A potential energy is always associated with a conservative force.
The choice U=0 is arbitrary, examples
Gravity: surface of the earth
Elastic: relaxed position
An object does not possess potential energy but a system does
Gravity: mass and the earth
Elastic: mass and the spring.
Electrical: the positive and negative charges.
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14. Conservation of Mechanical Energy
Let’s just consider a conservative system (earth and object, spring and mass…)
One in which the work does not depend on the path taken
Or equivalently one for which the work around a closed path is zero
One in which energy can be transformed from kinetic energy to potential and back again.
By the work-energy principle we know that
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15. What’s it Mean?
We have defined a conserved quantity, the total mechanical energy which is a constant. In symbols: K+U=constant .
Thus there can be well described transformation of energy from kinetic energy to potential energy. If K decreases then U must increase by an equivalent amount.
If only one object has kinetic energy then E=(1/2)mv2+U=constant. And for two positions: (1/2)mv12+U1=(1/2)mv22+U2
Here it’s clear why it doesn’t matter where we set the potential energy to zero. It would appear on both sides of the equation and cancel.
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16. The Classic Example: The Falling Rock
Anywhere along the path the total mechanical energy is E= (1/2)mv2+mgh.
When dropped, the rock has no velocity so K =(1/2)mv2=0 but it has potential energy U=mgh
As it falls, K increases and the potential energy decreases so that the total energy is constant.
At the surface, K is at a maximum and the potential at a minimum, U=mgh=0.
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17. What’s it good for?
If h=3.0m, calculate the speed when the rock is 1.0 m above the ground.
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19. Note this would have been very hard to solve with F=ma since the
Imagine the roller coaster starts at rest from the hill top at 40.0m.
What is:
The speed at the bottom of the hill?
The height at half this speed?
Let’s put y=0, U=0 at the bottom of the hill.
Note this would have been very hard
to solve with F=ma since the
vertical and horizontal motion
are coupled and curved.
Energy conservation avoided all that!
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20. What about Elastic Potential Energy?
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21. A Toy Dart Gun
A dart of mass 0.100kg is pressed against the spring of a toy gun. The spring (k=250N/m) is compressed 6.0cm and released.
What is the dart’s speed when leaves the spring at the relaxed position x=0?
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22. In the horizontal direction the only force (and potential energy) is due to the spring.
In the vertical direction gravity and the normal force cancel one another.
However after it leaves the barrel we are also equipped to describe the trajectory!
We can use the new equation with point 1 as corresponding to the compressed spring and point 2 the moment of departure.
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23. What about a Vertical Spring?
Then we have kinetic energy and two types of potential energy: gravitational and elastic.
The equation can be extended to:
And we can have energy transforming from kinetic energy to two forms of potential energy and back again…
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24. A Bungee Jumper.
A 75-kg student bungee jumps off a bridge. The cord has a spring constant of 50N/m starts to stretch at 15m.
How far will the student fall before coming to a stop? Ignore air-friction.
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26. Conclusions
Mechanical Energy and Conservation of Energy are very powerful concepts.
Friday we’ll discuss non-ideal situations, gravitational potential energy at all distances, power, and stabel and unstable equilibrium.
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