Solving Simple Harmonic Motion: Energy and Spring Calculations Explained

in this video we're going to focus on solving a problem that relates simple harmonic motion

with energy so we have a force of 300 newtons and it's used to stretch a horizontal

spring with a 0.5 kilogram block attached to it by 0.25 meters

the block is released from rest and undergoes simple harmonic motion so your task is to calculate the spring

constant the amplitude the maximum acceleration the mechanical energy of the system the maximum velocity and the

velocity when x is 0.15 so first let's understand what's happening in this problem so we have a

spring with a 0.5 kilogram mass

and let's say it's at its natural length so at this position the displacement is zero

now we're going to apply a force to stretch the spring and let's say it's over here now

so at this position it's at point 25 meters and to hold it at that position

requires a force of 300 newtons now once we let go of the spring that force will no longer be there so

the restoring force will cause the system to accelerate towards equilibrium and then it's going

to go backwards and forwards and it's going to oscillate back and forth so that's what's happening

in this system so knowing that how can we calculate the

value of the spring constant the spring constant k is equal to the force applied

divided by the change in life so it's f over x and it comes from hooke's law where f is

equal to kx so all we got to do is take the force of 300 newtons

and divided by the displacement which was 0.25 meters and so k

is 1200 newtons per meter so i'm just gonna write that up here

we're going to use that later so now let's move on to part b what is the amplitude

of this mass spring system the amplitude you need to know is the

maximum displacement basically is the maximum value of x now we stretch the spring by 0.25 meters

so the spring is going to oscillate between negative point 25 and 0.25

it doesn't have enough energy to go beyond that so x

can vary between negative a

and a so x could be anywhere between negative point 25

and 0.25 however a is a fixed value a is the maximum x value

so the amplitude is point 25 meters because x cannot be greater than that value

so that's it the amplitude is just the maximum displacement so in this example a

is 0.25 meters that's the answer to part b now let's move on to part c

calculate the maximum acceleration now this is a simple formula that can help us

get the answer the maximum acceleration is going to be the spring constant times

the amplitude divided by the mass so we have a spring constant of 1200 newtons per meter

and if we multiply it by .25 meters and then divide it by the mass which is

0.5 kilograms so in this example the maximum acceleration is 600

meters per second squared now newton's second law states that the net

force is mass times acceleration so you can also find the maximum acceleration by taking the force applied and dividing

by the mass of the system so the force applied was 300 newtons and the mass

is 0.5 kilograms so 300 divided by 0.5 will also give you an acceleration of

600 meters per second squared so if you have the force and mass you could find the acceleration or if you

have the spring constant the amplitude and the mass you can also find as well now let's move on to part d

calculate the mechanical energy of the system the mechanical energy is the total

energy of the system it's the sum of the kinetic and the potential energy of the system

but to find it here's the formula you need it's one half

multiplied by the spring constant times the square of the amplitude the spring constant is 1200

the amplitude is 0.25 so it's 0.5 times 1200 times 0.25 squared

and so the mechanical energy of the system is

37.5 joules part e what is the maximum velocity now before we

get the answer we need to talk about a few things so let's say the string is like fully

stretched at this point so let's say this is its equilibrium position

and here it's at point 25 and when it's fully compressed

x is negative 0.25 you need to know that the maximum velocity occurs when the

block is at the center basically at a position of

zero so that's where we have the maximum velocity

on the right side the velocity is zero on the left side it's zero now at the center the acceleration is

zero but at the edges the acceleration is at its maximum value

now the mechanical energy of the system is constant however the kinetic and potential energies vary at the center

because the block is moving at its greatest speed the kinetic energy

is at its maximum which means that all of the kinetic energy is equal to the total mechanical

energy which is 37.5 now at the edge when it's fully stretched

or fully compressed the potential energy is at a maximum so at those positions

the potential energy is equal to the mechanical energy but in the middle

the kinetic energy is equal to the mechanical energy but anywhere between let's say let's call

this point a b c

d and e so at points b and d or anywhere else other than a c and e

the mechanical energy is the sum of the kinetic and the potential energy but at point c the

mechanical energy is equal to the maximum kinetic energy and at point a and e the mechanical energy is equal to

the maximum potential energy so make sure you understand that so i'm just going to put that down in

writing you may want to write this down it's important for your tests

so at a you could set mechanical energy equal to potential energy and at e you could do the same thing

at point c you could set mechanical energy equal to the kinetic energy everywhere else let's say at point b and

d the mechanical energy is going to be the sum of the kinetic and the potential

energy so let's start with this formula mechanical energy is equal to kinetic

plus the potential energy now at part e the maximum velocity occurs when the

block is at its equilibrium position that is when x is equal to zero now we know that mechanical energy is

one half k times the amplitude squared and the kinetic energy is one half times mv squared

the potential energy of a spring is one half kx squared and at the center

when x is zero the potential energy is going to be zero so this term disappears

so that's why we can say that at the center mechanical energy is equal to the maximum kinetic energy

and that can help us to calculate the maximum velocity so we could cancel the one half

and so k is 1200

the amplitude is .25 and the mass

is 0.5 so 1200 times 0.25 squared divided by 0.5

is 150 and then take the square root of that answer and this will give you a maximum

velocity of 12.25 meters per second

now for those of you who want a simple equation to calculate the maximum velocity here

it is the maximum velocity is simply the square root of k divided by m

times the amplitude so if we type in the square root of 1200 divided by the mass of 0.5

multiplied by an amplitude of 0.25 that will give you the same answer 1200 divided by 0.5 is 2400

and the square root of 2400 times 0.25 that's going to give you this answer again 12.25 meters per

second now what about part f what is the velocity when x is .15 so how can we find the answer

the equation that we could use is this equation so if you want to find a velocity at any

position it's equal to the maximum velocity multiplied by 1

minus x squared over a squared so the maximum velocity is 12.25

and the current displacement is point 15 meters divided by the maximum displacement of

0.25 meters and don't forget to square it so 1 minus point 15 squared divided by

0.25 squared is 0.64 and if you take the square root of that that gives you 0.8 and then

multiply that by 12.25 so the velocity at a position of point 15 meters

is 9.8 meters per second now the other way to get the answer is

conservation of energy so at this position this would correlate to

point d which is between points c and e so if it's not at the center or at the

edge if it's anywhere in between you can set mechanical energy equal to kinetic

plus potential energy so mechanical energy is one half ka squared

the kinetic energy is one half mv squared and the elastic potential energy is one

half kx squared so let's plug in what we have k is 1200

the amplitude is 0.25 and then that's going to be one-half

times the mass of 0.5 and we're looking for v plus one-half times k which is

1200 times x which is point 15.

now we already know what this is half of 1200 is 600 and 600 times 0.25 squared that's

37.5 which is the mechanical energy that we had before half of one half is .25

and then half of 1200 which is 600 times point 15 squared that's going to be 13.5

so at this point we just got to solve for v so first let's subtract 37.5

by this number 13.5 and so that's going to be 24. so we have 24 is equal to 0.25

times v squared next divide 24 by 0.25

that will give you 96 and so 96 is equal to v squared and now we just got to take the square

root of both sides so the square root of 96 is 9.8 and so that will give us

that speed again so we get the same answer so as you can see

you could use conservation of energy to calculate the maximum velocity or the velocity at any position

or you could use the other equation to get that answer which was this one

so there's more than one way to solve these kinds of problems you