Question Video: Identifying a Physical Quantity by Its Dimensions Physics

Video Transcript

What is the physical quantity that has dimensions of 𝐿 times 𝑇 to the negative one? (A) Displacement, (B) velocity, (C) acceleration, (D) frequency, (E) angular frequency.

Looking at these dimensions, we know that 𝐿 represents length and 𝑇 represents time. Another way to write this is length divided by time. We want to identify which of these five physical quantities has these dimensions. If we consider first displacement, we know that displacement is a distance in a certain direction, sometimes represented by the letter 𝑠. Because displacement is a distance, the dimensions of 𝑠 are length. This is different from length divided by time. So we won’t choose answer option (A), and instead we’ll move on to option (B).

A velocity is a measure of displacement divided by time. Dimensionally then, velocity is a length divided by a time. We can see that these dimensions agree with the dimensions given to us in the problem statement. It looks then that option (B) will be our answer.

But just to make sure that it is, let’s continue on with option (C). Acceleration is defined as a change in velocity divided by a change in time. Therefore, the dimensions of acceleration are the dimensions of velocity divided by the dimensions of time. The dimensions of velocity can be written as 𝐿 times 𝑇 to the negative one. And those of time are simply 𝑇. If we multiply both numerator and denominator by 𝑇 to the negative one, then 𝑇 multiplied by 𝑇 to the negative one in the denominator cancels out. And we end up with dimensions of 𝐿 times 𝑇 to the negative two. These, we see, are not equal to 𝐿 times 𝑇 to the negative one. So we won’t choose answer option (C) either.

Moving on to option (D), frequency, the unit of frequency is the hertz. And this is equal to inverse seconds. If we have a frequency 𝑓 then, the dimensions of that frequency are one over time or inverse time. This also is different from the dimensions given to us in the problem statement.

Lastly, we consider answer option (E), angular frequency. If we have an angular frequency 𝜔, the units in which 𝜔 is normally written are radians per second. But, and here’s an important point, radians are dimensionless units. Therefore, when it comes to the dimensions of angular frequency, these are actually the same as the dimensions of frequency itself, an inverse time. This also is not a match for our given dimensions of length times time to the negative one.

For our answer, we choose option (B); velocity is the physical quantity that has dimensions of length times time to the negative one.