Displacement dimensional analysis?

Question

Suppose that the displacement of an object is related to the time according to the expression x=Bt^2. What are the dimensions of b?
also
A displacement is related to the time as x= Asin(2pi f t) where A and f are constants find the dimensions of A (hint: a trigonometric function appearing in an equation must be dimensionless)

Answer 1

Units, like meter, kilogram, second, can be treated like any variable in an equation. That is, we can multiply, subtract, add, and divide these units just like we do with variables.

In fact, we can set l = meter, m = kilogram, and t = second to show that our length metric (l) is a meter, our mass metric (m) is the kilogram, and the time metric (t) is the second. l, m, and t really are variables here. These are in fact, the standard so-called kg-m-s standard SI metrics.

So let's look at x =bt^2. x is length; so it's metric is l. t is time; so it's metric is t. To find the b metric (its dimensions), we solve for b = x/t^2; so that the metrics are b = l/t^2, which is length (l) per sec (t) squared. As a variable, t can be multiplied by itself to yield t^2. And we recognize b has the derived acceleration metric. [Note: x = bt^2 will not give x = 1/2 bt^2, where b is acceleration, because dimensional analysis does not reveal dimensionless factors like the 1/2 that completes the actual distance equation.]

Your second question is a slam dunk. Look at the hint. If the sine factor has to be dimensionless, then x = A sin() so that A = x/sin() and the dimensions (the metrics) are A = l/1 = l. When a factor is without dimensions (metrics), we simply put a dimensionless one (1) for that factor. Thus, the A metric is a length metric. [Note: kg-m-sec is just one of several SI standard dimensions. Another one is g-cm-sec, for gram, centimeter, second.]

Answer 2

Mass dont displace as an imaginary line . A mass displaces a volume of space.
The maximun distance that a mass travels during it displacement is equal to the product of its average acceleration and the time squared that the force is moving .
Therefore B in your equation is the average acceleration which has the dimensions of meters per seconds^2.
If the motion of the mass is analyzed in terms of a sinusoidal function; The A would be the maximum distance(in units of meters) that the force travel and 2pi f is the frequency(in units of seconds^-1) that the force moves during the mass displacement. The time of the total displacement of the mass is called the period of the oscillation.(measured in seconds).

Its vey vely simple.

Answer 3

x = Bt^2.

The dimensions of b is L T^-2
==================================
x= A sin(2pi f t)

sin(2pi f t) has no dimension

Hence A must have a dimension of lenght, L

=============================================

Answer 4

A displacement is a measure of distance(meters, feet, etc).
B and t would have to be either time or distance per time.
Most likely B is time.

I don't understand your second question.
It should still have dimensions. "t" is a scalar
of the frequency. Essentially, as time passes
the frequency increases.

Answer 5

First initiate with the instruments of acceleration. a = x / (t^2) be conscious that its the ratio of a distance divided by way of time squared. So, in case you multiply this set of instruments by way of time-squared, the consequence is a distance. Distance = x / (t^2) * (time)^2 Distance = x / (t^2) * (t)^2 Distance = x See it now? Its the comparable thought as in case you elevated velocity by way of time to get distance. velocity = x / t Distance = velocity * Time Distance = (x/t) * t Distance = x desire this helps, -Guru

Answer 6

Displacement will have units of length, being meters, feet, etc.
Time has units of sec,
so x[meters]=B[ ]*(t[sec])^2
then B[ ] = (x/t^2)[meters/sec^2]
So the units of B are
m/s^2