First post: work and energy problem

In summary, a 20.0 kg block attached to a horizontal spring of k = 2.0 kN/m is pulled to the right, extending the spring by 10.0 cm. Upon release, the block slides on a horizontal surface with a frictional force of 80.0 N. The maximum KE attained by the block as it slides to the point where the spring is unstretched is 3.6 J, which was found by applying the work-energy theorem and taking the derivative and setting it to zero to find the distance traveled by the block. The equation kx = Fd is dimensionally incorrect and should not be used to solve this problem.
  • #1
Ianardo
4
0

Homework Statement


A 20.0 kg block on a horizontal surface is attached to a horizontal spring of k = 2.0 kN/m. The block is pulled to the right so that the spring is extended 10.0 cm beyond its unstretched length, and the block is then released from rest. The frictional force between the sliding block and the surface has magnitude 80.0 N.

Homework Equations


What is the maximum KE attained by the block as it slides from release to the point where the spring is unstretched?

The Attempt at a Solution


KE is maximized when velocity is maximized. Therefore, the point is when the two forces are at equilibrium.
Set kx = Fd, d = (0.1-x)
2000(x) = 80(0.1-x)
x = 8/2080 (m)
= 0.003846 (m)

EPE (when fully stretched to 0.1 m) + Work done by friction = KE + EPE2
(.5)(2000)(0.1^2) - 80(0.1-0.003846) = .5(2000)(0.003846^2) + KE
Therefore, KE = 2.29 J, which makes sense.

I also did it the calculus way
W = the integral of force
F = ma, m =20 kg
a = [kx-f(0.1-x)]/20, integrating this yields velocity function, c =0
derive that function and set it to zero, yields the same answer.

But 2.29 J is not right. Why?
 
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  • #2
Ianardo said:
Set kx = Fd
What are you doing here? Compare the units on either side of this equation!

In general, always check that the units on each side of your equations match and never forget to write out the units.
 
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  • #3
Hi lanardo and welcome to PF.

Your starting equations are incorrect.
Ianardo said:
kx = Fd
This is dimensionally incorrect because kx has dimensions of [force] and Fd has dimensions of [Force×distance].
Ianardo said:
kx-f(0.1-x)
This is also incorrect for the same reason.
Do the problem by applying the work-energy theorem when the spring is at some distance x from its unstretched position and the mass is moving with speed v. Use symbols and put in the numbers at the very end so you can see what's going on.
 
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  • #4
I know what you mean. But how would you solve it if you were to not equate the two forces?
 
  • #5
Ok I see, thanks for the words!
 
  • #6
Got it! Thanks for both of your help!

3.6 J is the final answer. I used the work-energy theorem and took the derivative AND set it to zero, find x. Then plug the x back into solve for KE.

Thank you for your advice. Merry Christmas!
mass%20on%20spring%20j.jpg
 

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1. What is work and energy?

Work and energy are fundamental concepts in physics that describe the ability to do work or cause changes in an object's motion. Work is the product of a force applied to an object and the displacement of the object in the direction of the force. Energy is the ability to do work and can exist in many forms, including kinetic, potential, and thermal energy.

2. What is the relationship between work and energy?

The relationship between work and energy is described by the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. In other words, when work is done on an object, energy is transferred to or from the object, resulting in a change in its motion.

3. How do you calculate work and energy?

Work can be calculated by multiplying the force applied to an object by the distance the object moves in the direction of the force. Energy can be calculated using various formulas depending on the type of energy being considered. For example, kinetic energy can be calculated as 1/2 x mass x velocity squared.

4. Can work and energy be negative?

Yes, work and energy can be negative. This usually occurs when the direction of the force applied is opposite to the direction of the object's motion. In this case, work is considered to be done against the object, and the energy of the object decreases.

5. How is work and energy related to real-world problems?

Work and energy are essential concepts in understanding and solving real-world problems in various fields such as engineering, mechanics, and thermodynamics. For example, understanding the work and energy involved in a car's engine can help engineers design more efficient engines, and understanding the work and energy involved in a roller coaster can help ensure the safety of the ride.

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