A 10- -lb block A rests on a 40- lb plate B that is attached...

A $10-$ -lb block $A$ rests on a $40-$ lb plate $B$ that is attached to an unstretched spring with a constant of $k=60$ Ib/ft. Plate $B$ is slowly moved 2.4 in. to the left and released from rest. Assuming that block $A$ does not slip on the plate, determine ( $a$ ) the amplitude and frequency of the resulting motion, $(b)$ the corresponding smallest allowable value of the coefficient of static friction.

Key Concepts

Simple Harmonic Motion (SHM)

SHM refers to periodic motion where the net restoring force on an object is directly proportional to its displacement from the equilibrium position. This concept is fundamental in analyzing the oscillations of systems like a mass attached to a spring, where the amplitude and frequency determine the behavior of the motion.

Spring Constant and Hooke's Law

The spring constant is a measure of a spring's stiffness, indicating how much force is needed to stretch or compress the spring by a unit length. According to Hooke's Law, the force exerted by the spring is directly proportional to the displacement from the equilibrium position, which is a central idea in modeling oscillatory systems.

Mass and Weight Conversion

In mechanics, weight is the force due to gravity acting on an object while mass measures the amount of matter in the object. Converting weight to mass (using the acceleration due to gravity) is essential for applying Newton's laws of motion, particularly when analyzing the dynamics of systems involving oscillations.

Static Friction

Static friction is the force that prevents two surfaces from sliding past each other until a certain threshold is exceeded. The coefficient of static friction, which quantifies this threshold relative to the normal force, is crucial in situations where slipping must be avoided, ensuring that objects move together without relative motion between contact surfaces.

Transcript

00:01 In this question we are given a block on a plate that's attached via a spring to a wall and we are going to displace the plate to set the whole system into subharmonic motion and we're going to find the amplitude and frequency of that motion and then what's the smallest coefficient of friction that will keep block a from slipping on the plate? so let's first note to find this amplitude that this is initially an unstretched spring and then we're going to move plate b 2 .4 inches to the left.

00:44 So our amplitude a which you might also call xm, x max is what that stands for is going to be 2 .4 inches.

00:55 Since that's the furthest that we pulled our system from equilibrium that's the furthest that it can go.

01:02 And now to find the frequency we can find the frequency with the angular frequency divided by 2 pi and that angular frequency is given by the square root of k divided by m.

01:19 So 1 over 2 pi times the square root of k over m.

01:24 In order to do this we're going to need the mass of our system in slugs.

01:31 So we have a total of 40 pounds for the plate plus 10 pounds for the block so that's 50 pounds total divided by 32 .2 feet per second squared and i get a mass of 1 .553 slugs.

01:52 I do tend to keep an extra decimal place or two during calculations and then round at the end to improve the precision of that final answer.

02:03 So we now can take 1 over 2 pi times the square root of 60 pounds per foot divided by 1 .553 slugs and that gets me a frequency of 0 .989 hertz.

02:24 So we are just shy of one complete cycle per second.

02:30 So to find this corresponding smallest allowable value of the coefficient of static friction we need to consider that let's start like let's assume that we're going to start from the maximum displacement right here at the start...

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