Question

in a hypothetical situation 1 unit of mass is equal to 10 kg 1 unit of length = 5 M the unit of time equal to 2 second find 1 unit of force on the system solve by this n2=n1[m1/m2]×...

Answer 1

1 unit of force = 12.5 N

Answer 2

Explanation of the Solution:

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 ($L/T^2$). Therefore, the dimensional formula for force is $F = M \cdot L \cdot T^{-2}$.

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

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

• 1 unit of mass = 10 kg
• 1 unit of length = 5 m
• 1 unit of time = 2 s

To find the value of 1 unit of force in the new system, we substitute these equivalences into the dimensional formula for force:

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

Substitute the given values in SI units:

$1 \text{ unit of force} = (10 \text{ kg}) \times (5 \text{ m}) \times (2 \text{ s})^{-2}$

$1 \text{ unit of force} = (10 \text{ kg}) \times (5 \text{ m}) \times \frac{1}{(2 \text{ s})^2}$

$1 \text{ unit of force} = (10 \text{ kg}) \times (5 \text{ m}) \times \frac{1}{4 \text{ s}^2}$

$1 \text{ unit of force} = \frac{10 \times 5}{4} \text{ kg} \cdot \text{m/s}^2$

$1 \text{ unit of force} = \frac{50}{4} \text{ kg} \cdot \text{m/s}^2$

$1 \text{ unit of force} = 12.5 \text{ kg} \cdot \text{m/s}^2$

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

$1 \text{ unit of force} = 12.5 \text{ N}$

The method involving $n_2 = n_1 [M_1/M_2]^a [L_1/L_2]^b [T_1/T_2]^c$ leads to the same result. Here, we want to find $N_1$ (value in SI units) for $N_2 = 1$ (1 unit of force in the new system). $N_1 = N_2 \left[ \frac{M_2}{M_1} \right]^a \left[ \frac{L_2}{L_1} \right]^b \left[ \frac{T_2}{T_1} \right]^c$ For force, a=1, b=1, c=-2. $N_1 = 1 \times \left[ \frac{10 \text{ kg}}{1 \text{ kg}} \right]^1 \times \left[ \frac{5 \text{ m}}{1 \text{ m}} \right]^1 \times \left[ \frac{2 \text{ s}}{1 \text{ s}} \right]^{-2}$ $N_1 = 1 \times 10 \times 5 \times (2)^{-2}$ $N_1 = 10 \times 5 \times \frac{1}{4}$ $N_1 = \frac{50}{4} = 12.5$

Thus, 1 unit of force in the hypothetical system is 12.5 Newtons.