Question

Formula for true force is (A) \(\mathrm{F}=\mathrm{ma}\) (B) \(\mathrm{F}=[\\{\mathrm{d}(\mathrm{mv})\\} / \mathrm{dt}]\) (C) \(\mathrm{F}=\mathrm{m}(\mathrm{dv} / \mathrm{dt})\) (D) \(F=m\left(d^{2} x / d t^{2}\right)\)

Short answer

All four options represent the formula for true force: (A) \(F = ma\), (B) \(F = \frac{d(mv)}{dt}\), (C) \(F = m \frac{dv}{dt}\), and (D) \(F = m\frac{d^2x}{dt^2}\).

Step-by-step solution

Option A

\(F = ma\): This equation represents Newton's second law of motion. Here, F is the force, m is the mass, and a is the acceleration.

Option B

\( F = \frac{d(mv)}{dt} \): In this equation, F is the force, m is the mass, v is the velocity, and t is the time. This equation represents the rate of change of momentum with respect to time, which is mathematically equivalent to Newton's second law.

Option C

\(F = m \frac{dv}{dt}\): This equation represents the force acting on an object (F) as the product of mass (m) and the derivative of velocity (v) with respect to time (t). It is equivalent to Option A when we consider that the derivative of velocity with respect to time is equal to acceleration: \(a = \frac{dv}{dt}\). So, the equation becomes \(F=ma\).

Option D

\(F = m\frac{d^2x}{dt^2}\): Here, F is the force, m is the mass, x is the distance/displacement, and t is the time. This equation represents the force as the product of mass and the second derivative of position (distance) with respect to time, which is equivalent to acceleration. So, this equation is also a representation of Newton's second law - \(F=ma\). Conclusion: All four options represent the formula for true force in different ways, so the correct answer is (A) \(F = ma\), (B) \(F = \frac{d(mv)}{dt}\), (C) \(F = m \frac{dv}{dt}\), and (D) \(F = m\frac{d^2x}{dt^2}\).