If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

Options

A. `(aT)/(x)`
B. `at + 2 pi v`
C. `(aT)/(v)`
D. `aT + 4 pi^(2)v^(2)`

Answer

C

Explanation

The diminsions of given variables of SHM are as Displacement ` [x]=[M^(0) LT^(0) ] `
velocity ` " "[v] =[M^(0) LT^(-1) ] `
Accelertion, ` " " [a] =[M^(0) LT^(-2) ]`
and time period ` [T]= [M^(0) L^(0) T]`
Now, checking each option for these values For option (a)
` ([a][T] )/( [x] )=([M^(0) LT^(-2) ][M^(0) L^(0) T])/( [M^(0) LT^(0) ]) =[M^(0) L^(0) T^(-1) ]`
As it depends on time, so change with it
For option
` [a] [T]+2pi [v] =[M^(0) LT^(-2) ][M^(0) L^(0) T] +[M^(0) L^(0) T^(-1) ]= [ M^(0) LT^(-1) ] `
it is also dependent on time and hence changes with it.
.For option (c),
` ([a][T] )/( [v] ) =([M^(0) LT^(-2)] [M^(0) L^(0) T])/( [M^(0) LT^(_1) ] ) =[M^(0) L^(0) T^(0) ]`
As it a constant having no dimension, so it does not change with time,
For option (d)
` [a][t] +4pi^(2) [d]^(2) =[M^(0) LT^(-2)] [M^(0) L^(0)T]+ [M^(0) LT^(-1)]^(2) ` ltbr gt `= [LT^(-1) ]+ [L^(2) T^(-2) ]`
As,the term is dependent on time so changes with it.
Also, it is dimensionally incorrect.
Hence, option (c) is correct