Consider the equation
v = (1/4)zxt^2.
The dimensions of the variables v, x, and t are [L]/[T], [L], and [T], respectively. The numerical factor 4 is dimensionless. What must be the dimensions of the variable z, such that both sides of the equation have the same dimensions?
v = 1/4 zxt^2
z = 4v / xt^2
= (L/T) / (LT^2)
= 1/T^3
L/T = zLT^2
L/T รท LT = zT
1/T^2 = zT
1/T^3 = z
Well, this is quite a mathematical riddle! Let's put on our thinking caps and dive into it.
According to the given equation, the left side has dimensions of [L]/[T], which means the right side must also have the same dimensions for them to be equal.
We can break down the dimensions of the right side:
v = (1/4)zxt^2
[ L ] / [ T ] = (1/4) [ ? ] [ L ] [ T ]^2
To have the same dimensions on both sides, we need to figure out what goes inside those question marks.
From the given information, we know that [ L ] represents a length dimension and [ T ] represents a time dimension. Therefore, the question marks should denote what dimension z has.
Since [ L ] / [ T ] = (1/4) [ ? ] [ L ] [ T ]^2, we solve for [ ? ]:
[ ? ] = (1/4) / [ T ].
By simplifying, we can say:
[ ? ] = 1 / (4 [ T ]).
So, the dimensions of z, denoted by [ ? ], must be 1 / (4 [ T ]) to maintain the same dimensions on both sides of the equation.
Hope that helps!
To determine the dimensions of the variable z, we can set up a dimensional analysis of the equation.
Dimensions of v = [L]/[T]
Dimensions of x = [L]
Dimensions of t = [T]
Numerical factor 4 is dimensionless.
Using the equation v = (1/4)zxt^2, we can write the dimensional equation as follows:
[L]/[T] = (dimensionless)(L)(T^2)
To have the same dimensions on both sides, the dimensions of the left side should match the dimensions on the right side.
Comparing the dimensions, we can set up the following equation:
[L]/[T] = [L]*(T^2)
To equate the dimensions of [T] on both sides, we need to introduce a [T] term on the right side. We can do this by multiplying [L] by [T]^(-1).
So, the equation becomes:
[L]/[T] = [L]*[T]^{-1}*(T^2)
Simplifying further, we get:
[L]/[T] = [L]*(T^{2-1})*(T^0)
Simplifying the exponents, we get:
[L]/[T] = [L]*(T^1)*(T^0)
Again, equating the dimensions of [L] on both sides, we have:
[L]/[T] = [L]*[T]
Now, we can see that the dimensions are balanced. To ensure both sides have the same dimensions, the variable z should have the dimensions of [T].
Therefore, the dimensions of the variable z in the equation v = (1/4)zxt^2 should be [T].
To find the dimensions of the variable z, we need to equate the dimensions on both sides of the equation.
On the right-hand side of the equation, we have (1/4)zxt^2. Breaking it down, we can write it as:
(1/4)z * x * t^2
Now let's look at the dimensions of each term:
(1/4)z: This term has the same dimension as z, which we're trying to find.
x: The dimension of x is given as [L], representing length.
t^2: The dimension of t^2 is [T]^2, representing time squared.
Putting it all together, the dimensions of the right-hand side of the equation are:
(1/4)z * [L] * [T]^2
Now let's consider the left-hand side of the equation, which is represented by v. We're given that the dimensions of v are [L]/[T].
Equating the dimensions on both sides, we have:
[L]/[T] = (1/4)z * [L] * [T]^2
To make the dimensions on both sides of the equation the same, we must have:
(1/4)z * [L] * [T]^2 = [L]/[T]
Next, let's cancel out common dimensions:
(L on both sides) -> (1/4)z * [T]^2 = 1/[T]
Now, let's solve for z:
z = (4 * [T]) / [T]^2
Simplifying further:
z = 4 / [T]
Therefore, the dimensions of variable z must be [1]/[T].