If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities. Find the dimensions of energy.

The correct answer is [F] [A] [T²].

To find the dimensions of energy in terms of the new fundamental quantities force (F), acceleration (A), and time (T), we can use the method of dimensional analysis. First, let us recall the standard dimensions of these quantities in terms of mass (M), length (L), and time (T):

$\text{Energy}\left(E\right)=\left[M^1{L}^2{T}^{-2}\right]$

$\text{Force}\left(F\right)=\left[M^1{L}^1{T}^{-2}\right]$

$\text{Acceleration}\left(A\right)=\left[L^1{T}^{-2}\right]$

$\text{Time}\left(T\right)=\left[T^1\right]$

We can express the dimensions of energy as a product of the dimensions of force, acceleration, and time raised to unknown powers a, b, and c:

$\left[E\right]={\left[F\right]}^a{\left[A\right]}^b{\left[T\right]}^c$

Next, we substitute the standard dimensional formulas into the equation:

$\left[M^1{L}^2{T}^{-2}\right]={\left[M^1{L}^1{T}^{-2}\right]}^a{\left[L^1{T}^{-2}\right]}^b{\left[T^1\right]}^c$

Now, we expand the right side by multiplying the exponents:

$\left[M^1{L}^2{T}^{-2}\right]=\left[M^a{L}^{a+b}{T}^{-2a-2b+c}\right]$

According to the principle of dimensional homogeneity, the powers of corresponding base quantities on both sides of the equation must be equal. We will equate the powers of M, L, and T:

For Mass (M):
$1=a$
This immediately tells us that a = 1.

For Length (L):
$2=a+b$
Since we know a = 1, we can substitute it in to get 2 = 1 + b, which means b = 1.

For Time (T):
$-2=-2a-2b+c$
Substitute a = 1 and b = 1 into the equation:
$-2=-2\left(1\right)-2\left(1\right)+c$
$-2=-4+c$
$c=2$

Finally, we substitute the values of a, b, and c back into our original assumed expression for energy. With a = 1, b = 1, and c = 2, we get:

$\left[E\right]={\left[F\right]}^1{\left[A\right]}^1{\left[T\right]}^2$

This confirms that the dimensional formula for energy in terms of force, acceleration, and time is [F] [A] [T²]. Alternatively, this result can also be intuited from basic physics formulas: Work (Energy) is Force × Distance. Distance is dimensionally equivalent to Acceleration × Time² (from kinematics, d = 0.5 × a × t²). Therefore, Energy has dimensions of Force × Acceleration × Time².