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Chapter 3: Problem 329

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

Expert verified
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

01

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.
02

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.
03

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 \(F=ma\).
04

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) \(F = m\frac{d^2x}{dt^2}\).

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Most popular questions from this chapter

A body of mass \(5 \mathrm{~kg}\) starts from the origin with an initial velocity \(u^{\rightarrow}=30 \mathrm{i}+40 \mathrm{j} \mathrm{ms}^{-1}\). If a constant Force $\underline{F}=-\left(\mathrm{i}^{\wedge}+5 \mathrm{j}\right) \mathrm{N}$ acts on the body, the time in which the y-component of the velocity becomes zero is (A) \(5 \mathrm{~s}\) (B) \(20 \mathrm{~s}\) (C) \(40 \mathrm{~s}\) (D) \(80 \mathrm{~s}\)

A \(0.5 \mathrm{~kg}\) ball moving with a speed of \(12 \mathrm{~ms}^{-1}\) strikes a hard wall at an angle of \(30^{\circ}\) with the wall. It is reflected with the same speed and at the same angle. If the ball is in contact with the wall for \(0.25 \mathrm{~S}\) the average force acting on the wall is (A) \(96 \mathrm{~N}\) (B) \(48 \mathrm{~N}\) (C) \(24 \mathrm{~N}\) (D) \(12 \mathrm{~N}\)

With what acceleration (a) should a box descend so that a block of mass \(\mathrm{M}\) placed in it exerts a force \((\mathrm{Mg} / 4)\) on the floor of the box? (A) \((4 \mathrm{~g} / 3)\) (B) \((3 \mathrm{~g} / 4)\) (C) \(\mathrm{g} / 4\) (D) \(3 \mathrm{~g}\)

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