Question

If 'x', 'v' and 'a' denote the displacement, velocity and acceleration of a particle respectively executing SHM of periodic time t, then which one of the following does not change with time?

Options

• `"aT"/x`

• at + 2 π v

• `"aT"/"v"`

• aT + 4π²v²

Answer 1

`"aT"/"v"`

Explanation:

The dimensions of given variables of SHM are as Displacement, [x] = [M⁰ L T⁰]

Velocity, [v] = [M⁰ L T ^-1]

Acceleration, [a] = [M⁰ L T ^-2]

and time period, [T] = [M⁰ L⁰ T]

Now, checking each option for these values

For 1st option,

`(["a"]["T"])/([x]) = (["M"^0 "LT"^-2]["M"^0 "L"^0 "T"])/(["M"^0 "L" "T"^0]) = ["M"^0 "L"^0 "T"^-1]`

As it depends on time, so change with it.

For 2nd option,

[a][T] + 2π[v] = [M⁰ L T ^-2] [M⁰ L⁰ T] + [M⁰ L T ^-1]

= [M⁰ L T ^-1]

It is also dependent on time and hence changes with it.

For 3nd option,

`(["a"]["T"])/(["v"]) = (["M"^0 "LT"^-2]["M"^0 "L"^0 "T"])/(["M"^0 "L" "T"^-1]) = ["M"^0 "L"^0 "T"^0]`

As it is a constant having no dimension, so it does not change with time.

For 4th option,

[a][t] + 4π²[d]² = [M⁰ L T ^-2] [M⁰ L⁰ T] + [M⁰ L T ^-1]²

= [LT^-1] + [L²T^-2]

As, the term is dependent on time, so changes with it.

Also, it is dimensionally incorrect.

Hence, 3rd option is correct.