Question

A point oscillates along the `x` axis according to the law `x=a cos (omegat-pi//4)`. Draw the approximate plots
`(a)` of displacemetn `x`, velocity projection `upsilon_(x)`, and acceleration projection `w_(x)` as functions of time `t`,
`(b)` velocity projection `upsilon_(x)` and acceleration projection `w_(x)` as functions of the coordiniate `x`.

Answer 1

(a) Given`x=a cos (omegat=(pi)/(4))`
So, `v_(x)=x=-a omega sin (omegat-(pi)/(4))` and `w_(x)=ddot x=-a omega^(2)cos (omegat -(pi)/(4)) .......(1) `
On- the baiss of obtained expressions plots `x(t), v_(x)(t)` and `w_(x)(t)` can be drawn as shown in the answersheet, (of the problem book ).
`(b)` From Eqn (1)
`v_(x)=-a omega sin (omegat-(pi)/(4))` So,` v_(x)^(2)=a^(2)omega^(2)sin ^(2)(omegat-(pi)/(4)) ..............(2)`
But from the law `x=a cos(omegat-pi//4)`, so , `x^(2)=a^(2)cos^(2)(omegat-pi//4)`
or, `cos^(2)(omegat-pi//4)=x^(2)//a^(2)` or `sin^(2)(omegat-pi//4)1-(x^(2))/(a^(2)) .......(3)`
Using `(3)` in `(2)`,
`v_(x)^(2)=a^(2)omega^(2)(1-(x^(2))/(a^(2)))`or `v_(x)^(2)=omega^(2)(a^(2)-x^(2)) ......(4)`
Again from Eqn (4),`w_(x)=-a omega^(2)cos(omegat-pi//4)=-omega^(2)x`