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Finding the Potential Energy and Magnitude of Force

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1. If a certain spring stretches 8.93072 cm when a load of 18.6372 N is suspended from it, how much will the spring stretch if it is cut in half and when 30.2799 N is suspended from it? Answer in units of cm

2.A certain spring stretches 4.5 cm when it supports a mass of 0.77 kg. If the elastic limit is not reached, how far
will it stretch when it supports a mass of 10 kg? Answer in units of cm

3.In an arcade game, a 0.14 kg disk is shot across a frictionless horizontal surface by being compressed against a spring and then released. The spring has a spring constant of 241 N/m and is compressed from its equilibrium position by 7.8 cm. What is the magnitude of the spring force on the disk at the moment it is released?
Answer in units of N

4.A spring has a force constant of 517.5 N/m. Find the potential energy stored in the spring when the spring is
a) stretched 4.38 cm from equilibrium. Answer in units of J

5. (part 2 of 3) 10.0 points
b) compressed 2.35 cm from equilibrium. Answer in units of J

6. (part 3 of 3) 10.0 points
c) unstretched. Answer in units of J

7.A spring has a force constant of 80000 N/m. How far must it be stretched for its potential energy to be 21 J?
Answer in units of m

8. A spring with a force constant of 5.3 N/m has a relaxed length of 2.49 m. When a mass is attached to the end of the spring and allowed to come to rest, the vertical length of the spring is 3.69 m.
Calculate the elastic potential energy stored in the spring. Answer in units of J

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Finding potential energy of conservative force

See attached file for full problem description.

Previously it was considered a force of the form f = ixy + jcx^2 + kz^3, and found a value of "c" from the following list such that this was a conservative force.
Note: there must really be extra 'constants' in front of each term, with magnitude 1 but the proper units (such as Newton m^-2 for the first term) to make "F" have the right units of force. For this problem ignore these unit conversion constants and just use the right numerical magnitude of c, and these other constants.)
[A]: c = 1/4, [B]: c = 1/2
[C]: c = 1, [D]: c = 3

Now given the correct 'c', and using (x,y,z) = (0, 0,0) as the reference position, find the potential energy U(x, y, z) by computing the appropriate line integral from the reference position to the final (x, y, z).
Do the line integral even if you can see what the final U must be just by examing "F".

Finally, show that -deltaU provides the right F.

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