You are designing a delivery ramp for crates containing exercise equipment. The

Question

You are designing a delivery ramp for crates containing exercise equipment. The crates of weight 1490N will move with speed 1.8m/s at the top of a ramp that slopes downward at an angle 25.0?. The ramp will exert a 629N 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 7.9m along the ramp; this distance includes x. Once stopped, a crate must not rebound back up the ramp.

a- Calculate the maximum force constant of the spring

kmaxthat can be used in order to meet the design criteria.

Explanation / Answer

you want two constraints to be true.

First, the energy of the sliding crate be absorbed by the spring

.5*k*x^2=(1490*7.9*sin25)-(629*7.9)

and the force of the compressed spring be equal to the parallel component of the weight of the crate plus static friction
k*x=1490*sin25+629

solve for k

k=(1490*sin25+629)/x

x=2*(1490*7.9*sin25-629*7.9)/
(1490*sin25+629)

k=0.0043 N/m

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