Mass m = 45 kg is released from rest from the top of a rough...

Mass m = 45 kg is released from rest from the top of a rough ramp, with coefficient of kinetic friction 0.25 between the block and the incline, of height h = 3.2 m and length d = 5.5 m. At the bottom of the ramp, the mass slides on a horizontal, frictionless surface until it compresses a spring of spring constant k = 110 N/m. a. Calculate the speed of the mass at the bottom of the ramp? b. How far does the spring compress? c. How much work does gravity do on the mass when it slides down the ramp? d. How much work does the spring do on the mass during the compression? e. Calculate the work done by friction during the slide down the ramp. f. What is the total work done on the mass during the interval from the moment it is released to the moment the spring is fully compressed?

Transcript

00:01 In this example, i'm going to be looking at the concepts of work, force, and energy, and how they're all interrelated.

00:09 So we're looking at a mass resting on an inclined plane.

00:16 Mass equals 45 kilograms.

00:21 This plane has a height of, that's, h, 3 .2 meters, and a distance d of d equals 5 .5, means.

00:35 Meters.

00:37 All right.

00:38 And there's a coefficient of kinetic friction between the surface of the ramp and the block of 0 .25.

00:49 Okay, and now we're going to release this block from rest.

00:52 It's going to slide down this friction ramp and come to a zero friction horizontal surface, which has a spring attached to the end here.

01:07 So the block is going to collide with this spring, and we want to find the total distance that the spring is compressed.

01:17 And we know that the spring has a spring constant k of 110 newton's per meter.

01:25 So to do this, we'll need to find the final velocity of the block when it reaches the bottom of the ramp.

01:31 Right, and we can do this by using the work energy theorem that says the total sum of the work done on an object equals its change in kinetic energy.

01:44 So we're going to need to find out quite a few quantities here.

01:47 We're going to need to find the final velocity when the block reaches the end of the ramp.

01:53 And to do this, we're going to need the work done by gravity and the work done by friction as the block travels down the ramp.

02:02 That'll be our first step, and then we're going to need to know the work done by the spring as it compresses and brings the block to a rest.

02:12 All right, so the first thing we need to do is find our angle here.

02:19 The ramp is inclined at theta.

02:22 All right.

02:23 We know from trig sine of theta is going to equal opposite over hypotenuse.

02:30 So that's just my h over d, my 3 .2 meters over 5 .5 meters.

02:39 So theta equals 35 .6 degrees.

02:46 Now we need to find the work done by gravity and friction while the block is going down the ramp.

02:54 We know work done by gravity equals mgh as it goes from 3 .2 meters.

03:02 To zero ground level.

03:05 And so that equals my work, gravity.

03:09 Friction is a little bit trickier.

03:11 We know for a ramp the normal force.

03:13 Fn equals mg cosine theta.

03:20 All right, and we know work equals force times distance.

03:26 And we need to use the coefficient of friction as well to find our total work done by friction equals mu m .g.

03:37 Cosine theta times distance.

03:41 So times my 5 .5 meters.

03:44 And that's my work done by friction.

03:47 All right.

03:47 And using these two quantities and combine that with the conservation of mechanical energy, we have mgh minus my work of friction.

04:01 Remember, that's in the negative direction.

04:03 It always opposes the direction of motion.

04:05 That's going to equal my one -half mv squared...

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