You are designing a delivery ramp for crates containing exercise equipment. The 1470-N crates will move at 1.8 m/s at the top of a ramp that slopes downward at 22.0o. The ramp exerts a 515-N kinetic friction force 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 5.0 m along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the largest force constant of the spring that will be needed to meet the design criteria.

Solution 54P Step 1 of 4: Given data, Weight, W= 1470 N= mg Mass, m= (1470/g)=150 kg Friction force, f = 515 N k Speed of crate, v= 1.8 m/s Distance, d= 5 m To find, Compression, x= Force constant of spring, k= From work energy theorem, the energy stored in the spring will be equal to the sum of work done by gravity, friction and initial energy. That is, PE= W gravity W fric+ KE Using KE= mv ,PE= kx and W= F.d 2 2 1 2 1 2 2kx = W sin 22 .d f .k + mv2 ….1 When crate comes over the spring and rest, they exerts a force F which is due to weight and F=W For spring F= k x k=( x ) Using F=W k=( W )...........2 x