CP A child with poor table manners is sliding his 250-g dinner plate back and forth in SHM with

step 1 We know that the time period t equal to 2 pi into root of m upon k. So we are given information

CP A child with poor table manners is sliding his 250-g...

CP A child with poor table manners is sliding his $250-\mathrm{g}$ dinner plate back and forth in $\mathrm{SHM}$ with an amplitude of 0.100 $\mathrm{m}$ on a horizontal surface. At a point 0.060 $\mathrm{m}$ away from equilibrium, the speed of the plate is 0.400 $\mathrm{m} / \mathrm{s}$ . (a) What is the period? (b) What is the displacement when the speed is 0.160 $\mathrm{m} / \mathrm{s} ?$ (c) In the center of the dinner plate is a 10.0 -g carrot slice. If the carrot slice is just on the verge of slipping at the endpoint of the path, what is the coefficient of static friction between the carrot slice and the plate?

Key Concepts

Static Friction in Accelerated Systems

Static friction is the force that prevents relative motion between two surfaces in contact. In accelerating systems like an object placed on a moving oscillating plate, the maximum static friction must be sufficient to counteract the inertial force induced by acceleration. This concept allows one to calculate the minimum required coefficient of static friction to avoid slipping.

Energy Conservation in Simple Harmonic Motion

Energy conservation in SHM implies that the total mechanical energy, comprising kinetic energy and potential energy, remains constant throughout the motion. This principle leads to the relationship v² = ?²(A² - x²), which connects the speed of the oscillating object at any displacement with the amplitude and the angular frequency.

Simple Harmonic Motion (SHM)

SHM describes oscillatory motion in which the restoring force is directly proportional to the displacement and acts in the opposite direction, leading to sinusoidal motion. This concept forms the basis for analyzing systems where displacement, velocity, and acceleration vary predictably over time.

Angular Frequency and Period

Angular frequency (?) characterizes the rate of oscillation in SHM and is directly related to the period (T) by the relation T = 2?/?. This relationship is essential for linking the time domain of oscillation with its spatial characteristics, making it possible to determine the period of oscillation from measurable quantities like speed and displacement.

Transcript

00:01 We know that the time period t equal to 2 pi into root of m upon k.

00:08 So we are given information about vx at some particular x.

00:15 So the expression relating these two quantities come from the conservation of energy, conservation of energy.

00:25 So that is half m v square m v x square plus half k x square equal to half k a square.

00:40 So this gives you the conservation of energy.

00:43 So we can solve this equation for root of m by k and then use that result to calculate t.

00:52 Okay.

00:53 So what we are going to do is you have to be very careful with that.

00:57 First we are going to solve this equation for root of m upon k and now after solving this we are going to use this result to calculate t okay now m v x squared equal to k into a square minus x square okay it gives root of m upon k so root of m upon k equal to root of a square minus x square upon b x so we are putting the values root of bracket open 0 .100 meter d whole square minus 0 .060 meter d whole square minus 0 .060 meter t whole square upon 0 .440 meter per second.

02:09 So remember first we are going to find out the value of root of m upon k.

02:14 Okay.

02:15 So that is equal to 0 .2 .00 seconds.

02:22 Okay.

02:23 Now, when we have already found out m upon k, that is this.

02:29 Now we can find out t.

02:30 That is 2 pi root of m upon k that is 2 pi into 0 .2 .200 seconds that will be approximately equal to 1 .26 seconds.

02:50 So this is the answer for the a part.

02:54 This will be the answer for the a part of a question.

03:00 So this is our a part.

03:05 Now, let's move on to the b part of it.

03:09 Understood, root of m upon k we have found out.