Problem 17: A block of mass 5.5 kg is sitting on a...

Problem 17: A block of mass 5.5 kg is sitting on a frictionless ramp with a spring at the bottom that has a spring constant of 445 N/m (refer to the figure). The angle of the ramp with respect to the horizontal is 24°. 33% Part (a) The block, starting from rest, slides down the ramp a distance 46 cm before hitting the spring. How far, in centimeters, is the spring compressed as the block comes to momentary rest? 33% Part (b) After the block comes to rest, the spring pushes the block back up the ramp. How fast, in meters per second, is the block moving right after it comes off the spring? 33% Part (c) What is the change of the gravitational potential energy, in joules, between the original position of the block at the top of the ramp and the position of the block when the spring is fully compressed?

Transcript

00:01 Okay, so in this first question, we are asked to find out how far the spring is compressed.

00:06 We have a spring here as the block comes to momentary rest.

00:11 So we're going to basically be looking at how much energy does the block gain as it moves down the slope.

00:19 And then that will be converted to elastic potential energy in the spring.

00:24 So we'll say a here.

00:26 Now the energy gains, it's going to make the work done.

00:28 Is going to be force multiplied by distance and we're told that it moves 40, where is it, 46 centimeters.

00:39 So 46 centimeters for hitting the spring.

00:41 So what's the force going to be? well, the force is going to be the weight.

00:46 It's going to be 5 .5 multiplied by 9 .81.

00:50 However, this does not act downward, it acts down the slope.

00:54 To find the component acts down the slope, we know theta will be the angle to the normal.

01:00 Okay, the thesis is going to be about 24 degrees.

01:02 Therefore, the angle down the slope, to get the component the way down the slope, it's going to be the weight mg multiplied by sine 24, so sine theta.

01:10 So 5 .5 multiplied by 9 .81, sine 24, that's our force, and then multiply all of that by the distance, which is 0 .46 meters.

01:21 And that's going to get you to 10 .1 joules.

01:27 So by the time the block hits the spring it will have 10 .1 joules of kinetic energy.

01:33 So we can then use the equation e is equal to half kx squared for the energy stored in the spring.

01:40 Therefore the square root of two times the energy, so two times 10 .1 over 445, which the spring constant equals x and that is equal to 0 .21 meters or, 21 centimeters.

02:00 Then for part b here, so half block comes through spring, push the block back up the ramp, how fast mists per second is the block moving right after it comes off the spring? well, it's going to be the same speed as it was moving when it hit the spring, assuming energy's lost.

02:19 So we can say that 10 .1 joules is equal to the kinetic energy, which is equal to half mv squared.

02:29 Same idea.

02:30 V is equal to the square root of 2 times 10 .1 over the mass, which is 5 .5...

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