How Does Spring Compression Affect Motion on an Inclined Plane?

[sfram35]

Homework Statement

A spring (80 { N/m} has an equilibrium length of 1.00 { m}. The spring is compressed to a length of 0.50 {m} and a mass of 2.1 { kg} is placed at its free end on a frictionless slope which makes an angle of 41 degrees with respect to the horizontal. The spring is then released.
If the mass is not attached to the spring, how far up the slope will the mass move before coming to rest? If the mass is attached to the spring, how far up the slope will the mass move before coming to rest? Now the incline has a coefficient of kinetic friction uk. If the block, attached to the spring, is observed to stop just as it reaches the spring's equilibrium position, what is the coefficient of friction uk?

Homework Equations

PE(spring)=1/2kx^2
PE(gravitational)=mgh
KE=1/2mv^2

The Attempt at a Solution

I think you have to use conservation of energy where the equation would be:

mgh=1/2mv^2+1/2kx^2

 

[jbsteele]

For part 1:mgh = 1/2(2.1)(9.81)(0.50sin41) = 4.46= 1/2(2.1)(v^2)v = sqrt(8.91) = 2.98 s = vts = (2.98)(t) = 4.46 t = 1.49 secd = vtcos41 = (2.98)(1.49)(cos41) = 1.80 mFor part 2:mgh = 1/2(2.1)(9.81)(0.50sin41)+1/2(80)(0.50)^2 mgh = 8.93= 1/2(2.1)(v^2)+1/2(80)(x^2)v = sqrt(7.53) = 2.74 s = vts = (2.74)(t) = 8.93 t = 3.25 secd = vtcos41 = (2.74)(3.25)(cos41) = 2.89 mFor part 3:uk = Ffriction / Fn Ffriction = uk x Fn Ffriction = uk x mgcos41 Ffriction = uk x (2.1)(9.81)(cos41) Ffriction = 8.93 uk = 8.93 / (2.1)(9.81)(cos41) uk = 0.34

 

[tomcue]

To find the distance traveled, we can use the equation x= h/tanθ. However, since the spring is compressed and then released, the initial velocity is not zero, so we need to use the equation v^2=u^2+2as, where u is the initial velocity and a is the acceleration due to gravity (g). We can solve for u by setting the initial potential energy (mgh) equal to the initial kinetic energy (1/2mu^2), and then solving for u. Once we have u, we can plug it into the equation x= h/tanθ to find the distance traveled up the slope before the mass comes to rest.

If the mass is attached to the spring, the spring will act as a spring-mass system and we can use the equation PE(spring)=1/2kx^2, where x is the displacement of the mass from the equilibrium position. We can set this equal to the gravitational potential energy (mgh) and solve for x to find the distance traveled up the slope before the mass comes to rest.

To find the coefficient of friction, we can use the equation F=ukN, where F is the frictional force, N is the normal force (equal to the weight of the mass), and uk is the coefficient of kinetic friction. Since the mass stops at the equilibrium position, we know that the frictional force must be equal in magnitude to the spring force (kx), so we can set these two equal and solve for uk.

Overall, this problem involves using conservation of energy and Newton's laws of motion to determine the distance traveled and the coefficient of friction. It is important to carefully consider the initial conditions and use the appropriate equations to solve the problem.