Question

Consider the equation . The dimensions of the variables v, x, and t are , , and , respectively. The numerical factor 3 is dimensionless. What must be the dimensions of the variable z, such that both sides of the equation have the same dimensions?

Answer 1

The dimension of variable z must match the LT-1, L, or T dimensions of the other terms in the equation to maintain dimensional consistency.

Step-by-step explanation:

When considering the dimensional consistency of an equation, it is important to ensure that all terms on both sides of the equation have the same dimensional units. For instance, if we have the dimensions of variables v (velocity), x (position), and t (time) as LT-1, L, and T respectively, then each term in the equation should reflect these dimensions.

To find the dimension of the variable z so that it matches the dimensions of the other terms, we would need to compare it to the dimensions of known quantities. For example, if z represents a term in the equation involving velocity, then the dimension of z should also be LT-1. If it were a position over time term, the dimension of z would again be Time, or T. This ensures that all terms in the equation, including the numerical factor 3 which is dimensionless, are dimensionally consistent.

Answer 2

Part of the question is missing but here is the equation for the function;

Consider the equation v = (1/3)zxt2. The dimensions of the variables v, x, and t are [L/T], [L], and [T] respectively.

Answer = The dimension for z = 1/T3 i.e 1/ T - raised to power 3

Step-by-step explanation:

What is applied is the principle of dimensional homogenuity

From the equation V = (1/3)zxt2.

• V has a dimension of [L/T]

• x has a dimension of [L]

• t has a dimension of [T]
• from the equation, make z the subject of the relation
• z = v/xt2 where 1/3 is treated as a constant
• Substituting into the equation for z
• z = L/T / L x T2
• the dimension for z = 1/T3 i.e 1/ T - raised to power 3