Calculate the force of a spring

[parm12]

Introduction and Question:

You are designing a delivery ramp for crates containing exercise equipment. The crates weighing 1490 N will move at a speed of 1.80 m/s at the top of a ramp that slopes downward at an angle 21.0^\circ. The ramp exerts a kinetic friction force of 540 N on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 7.60 m along the ramp. Once stopped, a crate must not rebound back up the ramp.

Calculate the force constant of the spring that will be needed in order to meet the design criteria.

I am stuck on this only because I cannot extract from this data the distance the spring compresses. Can someone please offer some insight?

 

[Astronuc]

Possible one is to use two equations, with two unknowns, k and x.

One is the force equation and the spring force, F_sping = kx, and the other equation is an energy equation, in which the spring stored energy is E_sping=1/2 kx².

 

[kyle8921]

Hello! Based on the information given, we can calculate the force of the spring by using the conservation of energy principle. Since the crates are moving at a constant speed, we can assume that there is no change in kinetic energy and all of the potential energy at the top of the ramp is converted into elastic potential energy in the spring at the bottom of the ramp.

First, let's calculate the potential energy of the crates at the top of the ramp:
PE = mgh = (1490 N)(9.8 m/s^2)(sin 21.0^\circ)(7.60 m) = 20303.52 J

Next, we can calculate the maximum potential energy of the compressed spring:
PE = 1/2kx^2
where k is the force constant of the spring and x is the distance the spring compresses. Since the crates must not rebound back up the ramp, the maximum compression of the spring will occur when all of the potential energy is converted into elastic potential energy. Therefore, we can set the two potential energies equal to each other and solve for k:

20303.52 J = 1/2kx^2
k = (2)(20303.52 J)/x^2

Now we need to find the distance x that the spring compresses. We can use the work-energy theorem to find this distance:
W = Fd = (540 N)(7.60 m) = 4104 J
where W is the work done by kinetic friction force on the crates. This work is equal to the change in kinetic energy of the crates:
W = 1/2mv^2_f - 1/2mv^2_i
4104 J = 1/2(1490 N)(1.80 m/s)^2 - 1/2(1490 N)(0 m/s)^2
4104 J = 1215.6 J
Solving for x:
x = √(2W/k) = √(2(1215.6 J)/(k))

Now we can plug in this value for x into our equation for k:
k = (2)(20303.52 J)/[(√(2(1215.6 J)/k))^2]
k = 27.69 N/m

Therefore, the force constant of the spring needed to meet the design criteria is approximately 27.69 N/m. I