Max value of spring constant

In summary, when designing a delivery ramp for crates containing exercise equipment, the maximum force constant of the spring that can be used to meet the design criteria is 461.4 N/m. This is found by considering the kinetic and potential energy equations, as well as the forces of kinetic and static friction on each crate. Additionally, a free body diagram of the final state when the crate is at rest against the spring shows that there are two forces - the force of kinetic friction and the force of the spring - and since the crate is at rest, the sum of these forces is 0.
  • #1
Quincy
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Homework Statement


You are designing a delivery ramp for crates containing exercise equipment. The crates of weight 1500 N will move with speed 2.2 m/s at the top of a ramp that slopes downward at an angle 23.0 degrees. The ramp will exert a 586 N force of kinetic friction on each crate, and the maximum force of static friction also has this value. At the bottom of the ramp, each crate will come to rest after compressing a spring a distance x. Each crate will move a total distance of 7.6 m along the ramp; this distance includes x. Once stopped, a crate must not rebound back up the ramp. Calculate the maximum force constant of the spring k_max that can be used in order to meet the design criteria.

Homework Equations



F = -kx
PEs = 1/2 kx^2
PEg = mgh
KE = 1/2 mv^2

The Attempt at a Solution



1/2mv^2 + mgh = Ff*d + 1/2 kx^2
kx = 586 N

k = 586/x

1/2(1500/9.8)(2.2)^2 + (1500/9.8)(7.6sin(23)) = (586*7.6) + 1/2 kx^2

371.143 = 1/2(586/x)(x^2)
x = 1.27 m
k(1.27) = 586

k = 461.4 N/m -- this apparently is not the right answer; what am i doing wrong?
 
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  • #2
Draw a free body diagram of the final state when the crate is at rest against the spring. How many forces are there, and what are they from? Also, the sum of the forces will be 0 since everything is at rest.
 

What is the maximum value of spring constant?

The maximum value of spring constant depends on the material and design of the spring. It can range from a few hundred to a few thousand newtons per meter.

How is the maximum value of spring constant determined?

The maximum value of spring constant is typically determined through laboratory experiments, where the spring is stretched or compressed until it reaches its elastic limit. The resulting force and displacement data is then used to calculate the spring constant.

What factors affect the maximum value of spring constant?

The maximum value of spring constant is affected by the type of material used, the thickness and length of the spring, and the number of coils. The shape and design of the spring also play a role in determining its maximum value of spring constant.

What happens if the maximum value of spring constant is exceeded?

If the maximum value of spring constant is exceeded, the spring will no longer behave elastically and may permanently deform or break. This can lead to a loss of functionality for the spring and potential safety hazards.

Can the maximum value of spring constant be increased?

Yes, the maximum value of spring constant can be increased by using stronger and more durable materials, increasing the thickness and length of the spring, and optimizing its design. However, there will always be a limit to how much the maximum value of spring constant can be increased.

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