Work/energy - spring and inclined plane

In summary, an inclined plane of angle 20.0° with a spring force constant of 500 N/m is used to project a 2.50kg block at a speed of 0.750m/s towards the spring. The spring is compressed by a distance x when the block momentarily comes to rest. To solve for x, the equations for kinetic energy, elastic energy, and gravitational potential energy were used. A calculation error was found and corrected to determine the correct value for x.
  • #1
shawli
78
0

Homework Statement



An inclined plane of angle 20.0° has a spring force constant k = 500 N/m fastened securely at the bottom so that the spring is parallel to the surface (as shown in the attached photo). A block of mass m = 2.50kg is placed on the plane at a distance d = 0.300m from the spring. From this position, the block is projected towards the spring with speed v = 0.750m/s. By what distance is the spring compressed when the block momentarily comes to rest?

Homework Equations



K = 1/2 m * v2

Eelastic = Es = 1/2 k * x2

Eg = mgh

The Attempt at a Solution



I get a bit confused when handling all the different Energy/Work equations...
But here's what I did:
(my variable "x" is the distance that the spring displaces, and I tried to set my coordinate system so that the x-axis would be where the spring is at its compressed position but I'm not sure if I did it properly...)

Ugi + Usi + Ki = Ugf + Usf + Kf
(2.50)(9.80)(0.300 + x)sin20 + 0 + 1/2*2.50*0.7502 = 0 + 1/2 * 500 * x2 + 0

However when I solve for x, I get the wrong value.

I think I may have set up my equations for gravitational potential energy incorrectly ... Or maybe I set the whole thing up wrong heh.
 

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  • #2
Looks perfect to me! ...maybe your math is off...recheck your numbers...
 
  • #3
Bah, you're right, I had calculation errors -_-

But yay, that means I got it! Thanks for checking :)
 

Related to Work/energy - spring and inclined plane

1. What is the relationship between work and energy in a spring system?

In a spring system, work and energy are directly proportional. This means that the amount of work done on a spring is equal to the change in energy stored in the spring. When a force is applied to compress or stretch a spring, work is done and the spring gains potential energy. When the spring is released, the potential energy is converted back into kinetic energy.

2. How does the spring constant affect the work done on a spring?

The spring constant, represented by the symbol k, determines how much force is needed to stretch or compress a spring by a certain distance. The greater the spring constant, the stiffer the spring and the more work is required to change its length. This means that a higher spring constant will result in more energy being stored in the spring for a given displacement.

3. Can a spring system have potential energy without work being done on it?

Yes, a spring system can have potential energy without work being done on it. This can occur when the spring is already compressed or stretched, and therefore has stored potential energy, without any additional force being applied to it. In this case, no work is being done on the spring but it still has potential energy due to its displacement from equilibrium.

4. How does the slope of an inclined plane affect the amount of work done on an object?

The slope of an inclined plane, represented by the angle θ, affects the amount of work done on an object by changing the distance over which the object is moved. The greater the slope, the longer the distance and therefore the more work is required to move the object. This is because the component of the force acting parallel to the slope increases with a steeper angle, resulting in more work being done.

5. Can the work done on an object on an inclined plane be negative?

Yes, the work done on an object on an inclined plane can be negative. This occurs when the object is moving in the opposite direction of the applied force, such as when it is sliding down the incline due to gravity. In this case, the force and displacement are in opposite directions, resulting in a negative work value. This negative work represents energy being transferred out of the system, in this case, the object's kinetic energy being converted into heat due to friction.

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