You are designing a delivery ramp for crates containing...

You are designing a delivery ramp for crates containing exercise equipment. The 1550-N crates will move at 1.8 m/s at the top of a ramp that slopes downward at 22.0°. 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. Part A) Calculate the largest force constant of the spring that will be needed to meet the design criteria. Express your answer with the appropriate units.

Transcript

00:01 For this problem, we are trying to design a ramp and if i put this here and i have a spring here, of course, the weight w will act here, but i will have a normal force here and this normal force would definitely be equal to w cost theta.

00:31 I will have the sliding friction here opposing the sliding down of this of this block and i'll have another force here which i will call and not which i will call this force is actually w sin teta the red force.

00:58 Now if we want to find the spring constants for this spring in the first instance we want to find the total work done in the first instance we want to find the total work done in the this whole setup.

01:16 If we take the positive direction of this walk to be when this block is sliding down, you will see that when it hits this spring, the work done by the spring, we act opposite sliding down motion because the spring will try to move the block back, the restoring force of the spring.

01:35 So i will say that that is minus.

01:44 There will also be work done by this w -sign -teta as it is sliding down and because that is in the positive direction the sliding down direction i will say plus w sign theta multiplied by the distance into travel the frictional force is pointing opposite the sliding down direction so i will say its walk is also negative this is equal to w let me not use w for work let me just call okay let me say walk total let me call it in full therefore if i put in the variables minus half kx squared which is the work done on this done by the spring plus we're giving the weight there as 1550 newton we're giving the angle as 22 degrees we're giving the distance as five meters we're also giving the sliding friction as 515 this is equal to total work therefore the total work will be equal to minus 1 over 2 kx squared plus when i calculate that i will have 328 .2 let us keep this but from the work energy tearing we know that the work is simply going to be the change in the kinetic energy which is k e2 minus k ee or let me use final minus k e initial and so minus 1 over 2 k x squared plus 328 .2 is going to be equal to k e final minus k e initial when this block falls to the bottom it should come to rest what that means for us is that v final is zero and so i can simply say that um 320 8 .2 minus half kx squared is equal to what minus half what was the mass we're giving the weight so we can get the mass from there okay i will just write it here 1550 was the weight over 9 .81 that will be the mass we're giving b initial as 1 .8 meters per second and of course we're squaring that and so i can simply say from this that 328 .2 minus half kx squared is equal to minus 255 .96 when i solve this.

05:28 Therefore, half kx squared is equal to what when i take this one the other place, i will have 584 .1 .1.

05:40 Let me keep this equation...

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