Problem solving protocol
1. The question posed will be related to the energy an object has at some point in space. Take inventory of all the energy the object has at that point. Taking inventory involves asking these questions:
a) Does the object have any kinetic energy? (Is it moving?)
b) Does the object have any eleastic potential energy? (Is a spring stretched or compresed?)
c) Does the object have any gravitational potential energy? (Is the object above the place where h = 0?)

2. Now take the same inventory at some other place where you know the complete energy picture.

3. Apply the law of conservation of energy, setting inventory #1 = inventory #2.

Two circumstances can cause a variation in this theme: a) if more than one object is moving; or b) friction is at work. In either case one has to pose two qurdtions: where did the energy come from?" and "where did it go? These two quantitiies are equal.

1. A 5 kg block is raised a distance of 6 m. a) How much work did this take? b) What is the gravitational PE at this new location? c) If the block is dropped, what KE will it have as it hits the floor? d) How fast will it be moving at that time?Solution

2. A block slides down a friction-free inclined plane 1 m long when one end is raised .4 m higher than the other end. How fast is the block moving when it reaches the bottom? How fast is the block moving if a 2-m long plane is raised . 4 m?

3. A ball of mass m is attached to a string of length l. The string is tied to a support and the ball is free to swing. The ball is released from rest when the string is horizontal. a) How fast is the ball moving at the lowest point in its path? b) How fast is the ball moving when the string makes an angle of 30 degrees with the vertical? c) Calculate the tension on the string in each of the cases described above. Solution

4. A ball of mass m is attached to a string of length l. The string is tied to a support and the ball is free to swing.When the ball is in its lowest position, it is given a push, causing it to move through a vertical circle. a) How fast must the ball be moving at its highest point in order to keep the string taut? b) What minimum velocity at the bottom of the path must the ball have to ensure that it negotiates the circle?

5. A block whose mass is M1 rests on a friction-free table and is connected by a string to a second block M2 hanging over the table. The system is released. a) How fast are the blocks moving after M2 has fallen a distance h? Find a numerical answer if h =1 m; M1 = 1 kg; M2 = 4 kg? c) What if there were a coefficient of friction, u , between block 1 and the table. Solution

These are more difficult

6. A ball, mass M, is attached to one end of a string, length L; the other end of the string is fixed in place. The string is stretched taut in a horizontal position and the ball is released from rest at the 9 o’clock position. a)how fast is the ball moving at 6 o’clock? b) How fast is the ball movingat 7 o’clock? c) Find the tension in the string in each case described above.

7. Assume that it takes a certain amount of energy, E, to cause a car to move at 25 miles/hour. a) In terms of E, how much energy will it take to make the car move at 50 mi/hr? 75 mi/hr? b) In 1978, the maximum speed allowed on the Maine Turnpike was reduced from 70 to 55 mi/hr as an energy conservation measure. What fraction of the energy used as 70 mi/hr was saved by this maneuver? Solution

8. A block, m = 2.5 kg, slides on a friction-free track and is moving a 4 m/s at point at the top of a hill 6 m high. a) How fast is the block moving at the bottom of the hill? b) Now the block encounters a spring (k = 1000 N/m)at the bottom of the hill. By how much is the spring compressed?

9. A spring (k = 500 N/m) is compressed .4 m and a .3 kg ball is placed atop the spring. The spring is released, launching the ball. To what height does it rise?Solution

10. Same ball, same spring, tougher problem. The ball is dropped onto the spring from a height of 2.5 m. By how much is the spring compressed?

11. Go back to 8. Same block, same hill, same speed at the top. We remove the spring from the bottom of the hill and replace it with a high friction surface where mu = .3. How far does the block slide before coming to rest? Solution

POWER

12 A veteran mountaineer is planning an assault on Mt Everest.(altitude = 29028 ft = 8850 m). The mountaineer weighs in at 180lbs (use 82 kg). a How much energy will it take to get her to the top? b) If she gets her energy from donuts (gooey ones with chocolate frosting) and they have the energy content of 250 kcals each, how many donuts will she need to make it to the top? c) does this answer seem reasonable?

13. A shallow pond is .1 m in depth and 100 m^2 in area. Solar energy is incident on the pond at 1400 W/m^2. How long will it take before the average temperature of the pond rises 5 oC? Solution

1a = 294 J 1b = 294 J 1c = 294 J 1d = 10.8 m/s	2a = 2.8 m/s 2b = 2.8 m/s	3a = (2gL)^.5 3b = (2gLcos30)^.5	4a = (gL)^.5 4b = (5gL)^.5
5a = [(2 M2gh)/ (M1 + M2)]^.5 5b = 3.96 m/s	6a = (2gL)^.5 6b = (2gLcos30)^.5 6c = 3mgcos30	7a = 4E & 9E 7b = 38.3%	8a = 11.6 m/sec 8b = .58 m
9 = 13.6 m	10 = .18 m	11 = 22.7 m	delete
13 = 1496 sec = 24.94 min	14 = 6.79 = 7 donuts

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