Question

The magnitude of energy is 100 J. What will be its value if the units of mass and time are doubled and that of length is halved?
A. 100 New unit
B. 200 New unit
C. 400 New unit
D. 800 New unit

Answer 1

Hint
Energy of a body is dependent on the mass, length and time. When these units are changed, the change in the value of the energy can be determined by performing dimensional analysis.
The dimensional equation for energy is given as: $E = M{L^2}{T^{ - 2}}$, where M, L and T are the base units of mass, length and time, respectively.

Complete step by step answer
We have the value of energy in standard units, and are asked to find the new value when the basic units change. For this, we first see the dependence of energy on mass, length and time as:
$\Rightarrow E = M{L^2}{T^{ - 2}}$ [Eq. 1]
We are given that these units change as:
Mass $M = 2M$ [Doubled]
Length $L = 0.5L$ [Halved]
Time $T = 2T$ [Doubled]
Energy $E = 100J$
On substituting these changes in the Eq.1, we get the new value of energy as:
$\Rightarrow E = 2M \times {[0.5L]^2}{[2T]^{ - 2}}$
On opening the brackets, we get:
$\Rightarrow E = \dfrac{{2M \times 0.25{L^2}}}{{4{T^2}}} = 0.125M{L^2}{T^{ - 2}}$
Hence, the unit of energy will be given as:
$\Rightarrow \dfrac{E}{{0.125}} = M{L^2}{T^{ - 2}}$
$\Rightarrow {E_{new}} = M{L^2}{T^{ - 2}} = 8E = 800$
Hence, the correct answer is option (D): 800 J.

Note
As we saw, dimensional analysis is helpful in determining the new units and values of a physical quantity when we know the changes in its base parameters. The quantity may also depend on a constant of proportionality whose values are determined experimentally. Similarly, for dimensionless constants, this method does not work. We also cannot use this procedure when exponents and trigonometric functions are involved.