Question

Density (p) of a body depends on the force applied (F), its speed (v) and time of motion (t) by the relation p = KF^av^bt^c , where K is a dimensionless constant. Then

• A. a=1, b=-4 and c=-2
• B. a=2, b=-4 and c=-1
• C. a=-1, b=-4 and c=2
• D. a=1, b=4 and c=-2

Answer 1

The Correct Option is A

The density of a body can be expressed as:
p = KF^a * v^b * t^c
where K is a dimensionless constant and a, b, and c are constants.
We can determine the values of a, b, and c by considering the units of the given parameters. The SI units of force, speed, and time are newtons (N), meters per second (\(\frac{m}{s}\)), and seconds (s), respectively. The SI unit of density is kilograms per cubic meter (\(\frac{Kg}{m^3}\)).
From the given relation, we can express K in terms of the units of the parameters:
p = KF^a * v^b * t^c
K = \(\frac{p}{ (F^a * v^b * t^c)}\)
Substituting the units of the parameters, we get:
K = \(\frac{(\frac{kg}{m^3}) }{(N^a * (\frac{m}{s})^b * s^c)}\)
We can simplify this expression by rearranging the units using the rules of exponents:
K = \((\frac{kg}{ N^a * m^b * s^c})^\frac{1}{m}\)
Comparing the units of the expression inside the parentheses with the units of the given parameters, we can determine the values of a, b, and c:
a = 1 (force has units of N)
b = -4 (speed has units of \(\frac{m}{s}\), which cancels out the length unit of meters in the numerator of K)
c = -2 (time has units of \(s^1\), which cancels out the \(s^c\) term in the denominator of K)
Therefore, the correct answer is a = 1, b = -4, and c = -2. (Option A).
Answer. A