A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible

A thrill-seeking cat with mass 4.00 kg is attached by a...

Key Concepts

Simple Harmonic Motion (SHM)

SHM describes periodic motion in which the restoring force is directly proportional to the displacement from equilibrium and oppositely directed. In vertical oscillators, this means that the combination of gravitational and elastic forces leads to a smooth back-and-forth motion around an equilibrium position.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring or elastic material when it is stretched or compressed from its natural, unstressed length. It is given by the expression ½kx², where k is the spring constant and x is the displacement. This energy plays a central role in systems where elastic forces are significant, such as oscillating springs.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, quantified by the formula ½mv², where m is the mass and v is the velocity. In oscillatory systems, the kinetic energy varies throughout the cycle, becoming maximal when the object passes through the equilibrium position.

Gravitational Potential Energy

Gravitational potential energy is the energy associated with an object's position in a gravitational field, typically calculated as mgh, where m is mass, g is the acceleration due to gravity, and h is the height relative to a chosen reference level. It is an important component in analyzing vertical motions where height changes are involved.

Energy Conservation in Oscillatory Systems

Energy conservation in oscillatory systems implies that the sum of the kinetic, elastic potential, and gravitational potential energies remains constant in the absence of non-conservative forces like friction or air resistance. This principle allows us to understand and predict how energy shifts between various forms as the system oscillates.

Transcript

00:01 In this question, we are calculating the elastic potential energy, gravitational potential energy, kinetic energy, and total energy for an object that is oscillating vertically on a spring.

00:18 So we are going to for simplicity, depict this scenario as a box.

00:33 And i'm not going to.

00:34 Well, yeah, we'll draw it as a spring.

00:41 So, we are told, actually need to draw that a lot bigger.

00:44 So we are told that the highest position of the oscillation, nearly enough coils, the highest position of this oscillation for this box is the unstretched length of the spring.

01:09 And that's, you know, that box was supposed to be doesn't matter if we measure the box or the spring, because that little bit is constant for all of it so this is now, we're going to call it x equals zero, right, because this is where we have no stretch.

01:29 And then, just for convenience i'm going to label the bottom of our stretch we're going to pretend this thing could go all the way down to the bottom of my table here.

01:41 And so this is going to be where we have a height, equal to zero.

01:46 We are told that there is an amplitude of oscillation, equal to 0 .05 meters.

01:56 So, this middle dashed line that i just drew is the equilibrium position which i've labeled eq.

02:03 And so, our object our cat is oscillating with amplitude a around that position.

02:14 And so, this lets us begin to see that at x equals zero the unstretched length of the spring, our object is elevated a height of 2a above its lowest possible point.

02:29 And then we also are going to know that at the highest point and the lowest point, our kinetic energy is going to be zero.

02:38 And we're starting to now form a picture of how to approach this question, once we have this diagram to help us visualize everything that's going on.

02:47 So, at the highest point, our elastic, so we know that elastic potential energy is calculated with one half k x squared.

03:00 But since x is zero, we know that this is going to be zero joules.

03:04 And similarly, we know that when our cat is at the highest point of its bounce, the kinetic energy, which is one half mv squared is going to be zero joules again, because v is zero at the top.

03:27 And i'm going to go ahead and go back and add over here because x is zero meters.

03:33 And so, this tells us then that all of the energy stored in this system is going to be from the gravitational potential energy.

03:41 And we can calculate gravitational potential energy with mgh.

03:46 And like we discussed when making our diagram, that means we need mg times twice that amplitude.

03:54 So we'll take our 4 .0 kilograms times 9 .8 times 2 times 0 .05 meters for that amplitude.

04:05 And that gives me a gravitational potential energy of 3 .92 joules.

04:11 Oops, that 9 is a little bit close to my decimal point.

04:12 There we go.

04:14 Now it looks like a 7.

04:16 Let's try this again.

04:17 There we go.

04:18 3 .92 joules.

04:19 And so our total over here is going to be zero plus 3 .92 plus zero, which is going to be 3 .92 joules.

04:32 And now because this is a conservation of energy problem, hey, that means our total energy in this system is going to be 3 .92 joules for all three positions.

04:49 Because of coe, conservation of energy.

04:56 Okay, so now let's take a look at that lowest point.

05:01 Once again, we now know that our kinetic energy is zero joules for the same reason as before, that the cat's not moving.

05:14 And now we're going to know that our gravitational potential energy is zero joules...

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