Question

(b) A distance is related to time according to the expression x = A sin(2πft), where A and f are constants. Find the dimensions of A. Again, "L" is the length dimension and "T" is the time dimension. [Hint: A trigonometric function appearing in an equation must be dimensionless.]

Answer 1

A is in length dimensions

Explanation:

The expression:

x = A sin (2πft)

has in the second member two factors A and sin (2πft); a sine is a relation between two sides with the same dimension that means a sine is a number ( with minimum and maximum values of 0 for zero degrees and 1 for 90 degrees ). As t is in units of time ( seconds, minutes or hours) frequency "f", which is the number of cycles per unit of time ( seconds, minutes or hours), t and f should be both in the same unit, in order to get just a number for sin2πf.

Therefore A should be in units of length and x will get its units from A

For instance

x = A sin(2πft)

t in seconds f in 1/seconds A in meters

By substitution, we can see that

x[ m ] = A [m] * sin[ 2π*sec* 1/sec ]

x[ m ] = A [m] * number