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Rotational kinetic energy and spring

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24. A 2.5-kg ball and a 5.0-kg ball have an elastic collision. Before the collision, the 2.5-kg ball was at rest and the other ball had a speed of 3.5 m/s. What is the kinetic energy of the 2.5-kg ball after the collision?
25. A 50-kg rider on a moped of mass 75 kg is travelling with a speed of 20 m/s. Each of the two wheels of the moped has a radius of 0.2 m and a moment of inertia of 0.2 kg.m2. What is the total rotational kinetic energy of the wheels?
26. A 1.0-kg wheel in the form a solid disk rolls along a horizontal surface with a speed of 6.0 m/s. What is the total kinetic energy of the wheel?
27. A 2.0-kg solid cylinder of radius 0.5 m rotates at a rate of 40 rad/s about its cylindrical axis. What power is required to bring the cylinder to rest in 10 s?
28. A solid cylinder of radius 0.35 m is released from rest from a height of 1.8 m and rolls down the incline as shown. What is the angular speed of the cylinder when it reaches the horizontal surface?
29. A ceiling fan has five blades, each with a mass of 0.34 kg and a length of 0.66 m. The fan is operating in its 'low' setting at which the angular speed is 9.4 rad/s. If the blades can be approximated as uniform thin rods that rotate about one end, what is the total rotational kinetic energy of the five blades?
30. A spring required a force of 1.0 N to compress it 0.1 m. How much work is required to stretch the spring 0.4 m?
31. A spring with constant k = 40.0 N/m is at the base a frictionless, 30.0 degrees inclined plane. A 0.50-kg block is pressed against the spring, compressing it 0.20 m from its equilibrium position. The block is then released. If the block is not attached to the spring, how far along the incline will it travel before it stops?
32. A ping-pong ball weighs 2.5*10^-2 N. The ball is placed inside a cup that sits on top of a vertical spring. If the spring is compressed 0.055 m and released, the maximum height above the compressed position that the ball reaches is 2.84 m. Neglect air resistance and determine the spring constant.
33. The spring constant for the spring in a special cannon is 1800 N/m. In cocking the cannon, the spring is compressed 0.55 m. What is the initial speed of a 7.0 kg cannon ball at rest on the free end of spring when it released?

https://brainmass.com/physics/conservation-of-energy/rotational-kinetic-energy-spring-174902

SOLUTION This solution is FREE courtesy of BrainMass!

24. For an elastic collision, both the energy and momentum are conserved.
Conservation of momentum:

Conservation of energy:

Solve the two equations, v1 = 14/3 m/s, v2 = 7/6 m/s
So the kinetic energy of the 2.5 kg ball is

25. The rotational kinetic energy of one wheel is

So the two wheels is 2 * 1000 = 2000 J.

26. The translational kinetic energy is
The moment of inertia of the solid disk is
So the rotational kinetic energy is
Therefore, the total kinetic energy is 18 + 9 = 27 J.

27. The rotational kinetic energy is

Pt = 200
P * 10 = 200
P = 20 W

28. when at the top of the slop, the total energy is mgh.
At the horizontal surface, the total energy is only the kinetic energy.
By conservation of energy:

29. The rotational kinetic energy of one blade is

5 * 2.18 = 11 J.

30. By Hook's law, F = kx
1.0 = K * 0.1
So the spring constant is k = 10 N/s
The work done is equal to the change of the elastic potential energy

31. Assume the height where the block stops is h.
When the spring is compressed, the elastic potential energy is 0.5kx2.
After the block stops, the total potential energy is mgh
By conservation of energy

Then the distance along the incline is

32. Using the conservation of energy, elastic potential energy to the gradational potential energy: