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 1520 N will move with speed 2.0 m/s at the top of a ramp that slopes downward at an angle 24.0 . The ramp will exert a 618 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 8.0 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 kmax that can be used in order to meet the design criteria.

Explanation / Answer

here,

distance along the ramp , d = 8 m

initial speed , u = 2 m/s

weight of crate , w = 1520 N

mass , m = w/g = 155.1 kg

kinetic friction force , ff = 618 N

let the spring constant be k

using work energy theorm

work done by friction force = initial kinetic energy + potential energy of crate - potential energy of spring

ff * d = 0.5 * m * u^2 + m * g * ( d * sin(theta)) - 0.5 * k * x^2

618 * 8 = 0.5 * 155.1 * 2^2 + 155.1 * 9.8 * ( 8 * sin(24)) - 0.5 * k * x^2

k = 624.1 /x^2

the maximum force constant of the spring is 624.1 /x^2

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