Potential Energy, Springs, and Friction

[cowmoo32]

Homework Statement

A block of mass m rests on a plane inclined at an angle θ with the horizontal. A spring with force constant k is attached to the block. The coefficient of static friction between the block and plane is μ_s. The spring is pulled upward along the plane very slowly.

What is the extension of the spring the instant the block begins to move. (Use any variable or symbol stated above along with the following as necessary: g.)

Homework Equations

F = ma
F = -kx

The Attempt at a Solution

I figured I can sum up forces in X and Y, with X being in the direction of the plane, and solve for x, but it's coming back as incorrect. I tried reversing the signs in case I missed a negative as well. The only other option I can see is summing the forces in the X direction equal to zero which would remove the ma term, but the system could be in equilibrium for any value of X and the block still not move. My thoughts are that setting Fx=ma finds x the instant it moves. I only have one submission left so I wanted to check before I go any farther.

ƩF_y=F_N-mgcos(θ)=0

F_N=mgcos(θ)ƩF_x=-F_Nμ_s-mgsin(θ)+kx=ma

Substitute F_N
-mgcos(θ)μ_s-mgsin(θ)+kx=ma

Solving for x
(-mgcos(θ)μ_s-mgsin(θ)-ma)/k=x

 

[cepheid]

At the instant the block begins to move, the spring force pulling up the incline has *only just barely* exceeded the weight and the static frictional force that are pulling down the incline. So you can set sum of the forces = 0 in the horizontal direction to get the critical value of x above which the thing will begin moving.

In other words, do ƩF_x = 0, NOT ƩF_x = ma, because what the heck would you use for the value of a?

 

[cowmoo32]

Gotcha, thanks. And just to be clear, is the spring -kx or kx since it's in the positive x direction? Negatives kill me.

Edit: Never mind. If I make it negative, I get a negative value for x, which makes no sense given my origin.

 

[cepheid]

Gotcha, thanks. And just to be clear, is the spring -kx or kx since it's in the positive x direction? Negatives kill me

The - sign in F = -kx refers to the fact that the force always opposes the spring displacement. Stretch the spring, and the force tries to compress it back. Compress the spring, and the force tries to stretch it.

However, in this case, you've chosen a coordinate system in which the positive x-direction is "up the plane". So, since the spring is being stretched, the restoring force wants to compress it again, which corresponds to pulling the mass *up the plane.* Hence, the force *on the mass* is positive, with magnitude kx.

 

[Nickie]

x=(-mgcos(θ)μs-mgsin(θ)-ma)/k

This solution seems correct. However, it is important to note that this equation only gives the extension of the spring at the instant the block begins to move. To find the extension at any other point during the motion, you would need to also take into account the acceleration of the block and the changing forces acting on it. Additionally, the direction of the spring force should also be considered, as it may change direction as the block moves. Overall, this is a good start to solving the problem, but further analysis may be needed to fully understand the behavior of the block and spring system.