The motion of a vibrating system is described by the function y(x, t) = A0e^(-at)sin(kx-ωt). If t refers to time in seconds and x refers to distance in meters, what must the units of k be?
1/s
s
1/m
unitless
m/s
m
k x must be the same units as ω t which is 2 pi f t which is 2 pi t/T which is no units (angle) so
k x must be no units so k must be 1/meter
by the way if L is wavelength and T is period
sin(kx-ωt) is the same as sin (2 pi x/L - 2 pi t/T)
so kx has same units as x/L so k has same units as 1/wavelength
Well, isn't this function just vibin' its way through the world of physics? So, let's break it down and find out what units k must have.
In the given function, kx represents a distance, which is measured in meters. Meanwhile, t refers to time, measured in seconds. Since sin(kx - ωt) is a unitless function, we can ignore it for now.
So, we have kx with units of meters. To cancel out the meters, we need the other term, k, to have units of 1/meter or 1/m.
Therefore, the units of k must be 1/m or "inverse meter"! Now, don't go running around telling people that you've got units of k-measurement figured out. They might think you're trying to measure their clown shoes!
The units of k must be 1/m.
This can be determined by analyzing the given equation: y(x, t) = A0e^(-at)sin(kx-ωt).
The term sin(kx-ωt) represents the spatial variation of the vibrating system, where k and x have the same units. Since the units of x are in meters, the units of k must be the reciprocal of x, which is 1/meter or 1/m.
To determine the units of the variable k in the given function, we need to analyze its role within the equation and consider the dimensions of the other quantities involved.
In the given equation: y(x, t) = A0e^(-at)sin(kx - ωt)
We can see that k appears as the coefficient of x in the sine term. Since the argument of the sine function must always be dimensionless, we can infer that the units of k must be reciprocal to the units of x in order for the product kx to be dimensionless.
Given that x is measured in meters (m), the units of k must be reciprocal to meters (1/m) in order to yield a dimensionless quantity.
Therefore, the units of k must be 1/m.