Question

if one unit of mass equal to 20 kg 1 unit of length equal to 4 m per unit of time equal to 2 second find one unit of force on the system

Answer 1

20 N

Answer 2

To find one unit of force in the new system, we first need to recall the fundamental definition of force.

1. Definition of Force:

Force (F) is defined by Newton's second law as the product of mass (M) and acceleration (A). Acceleration is length (L) divided by time (T) squared ($L/T^2$).
So, the dimensional formula for force is:
$F = M \cdot A = M \cdot \frac{L}{T^2}$

In the SI system, 1 Newton (N) is defined as the force required to accelerate a mass of 1 kg by 1 m/s².
Therefore, $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$.

2. Given Units in the New System:

We are given the following relationships between the new system's units and SI units:

• 1 unit of mass ($M_{\text{new}}$) = 20 kg
• 1 unit of length ($L_{\text{new}}$) = 4 m
• 1 unit of time ($T_{\text{new}}$) = 2 s

3. Calculate One Unit of Force in the New System:

Let 1 unit of force in the new system be $F_{\text{new}}$. Using the dimensional formula for force, we can express 1 unit of force in terms of the given new units:

$1 \text{ unit of force} = (1 \text{ unit of mass}) \times \frac{(1 \text{ unit of length})}{(1 \text{ unit of time})^2}$

Now, substitute the equivalent SI values for each new unit:

$1 \text{ unit of force} = (20 \text{ kg}) \times \frac{(4 \text{ m})}{(2 \text{ s})^2}$

$1 \text{ unit of force} = (20 \text{ kg}) \times \frac{(4 \text{ m})}{(4 \text{ s}^2)}$

$1 \text{ unit of force} = (20 \times \frac{4}{4}) \text{ kg} \cdot \text{m/s}^2$

$1 \text{ unit of force} = 20 \text{ kg} \cdot \text{m/s}^2$

Since $1 \text{ kg} \cdot \text{m/s}^2 = 1 \text{ Newton (N)}$, we have:

$1 \text{ unit of force} = 20 \text{ N}$

Thus, one unit of force in the given system is equal to 20 Newtons.

Explanation of the solution:

Force is dimensionally $M L T^{-2}$. Substitute the given values of 1 unit of mass (20 kg), 1 unit of length (4 m), and 1 unit of time (2 s) into this dimensional formula.
$1 \text{ unit of force} = (20 \text{ kg}) \times (4 \text{ m}) \times (2 \text{ s})^{-2} = 20 \times 4 \times \frac{1}{4} \text{ kg} \cdot \text{m/s}^2 = 20 \text{ N}$.