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Chapter 12: Problem 4

Write Newton's Second Law of Motion for three-dimensional motion with only the gravitational force (acting in the \(z\) -direction).

Short Answer

Expert verified
Answer: The expression for the acceleration vector in three-dimensional motion when only the gravitational force is acting in the z-direction is: \(\vec{a} = 0\hat{i}+0\hat{j}+(-g)\hat{k}\), where \(g\) is the acceleration due to gravity, approximately equal to \(9.81\,\text{m/s}^2\).

Step by step solution

01

Review Newton's Second Law of Motion

Newton's Second Law of Motion states that the acceleration \(\vec{a}\) of an object is directly proportional to the net force \(\vec{F}\) acting on it and inversely proportional to its mass \(m\). Mathematically, this is written as: \[\vec{F}=m\vec{a}\]
02

Break Down Three-Dimensional Motion

For a three-dimensional motion, the net force acting on an object can be broken down into components in the x, y, and z directions: \[\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}\] Similarly, the acceleration vector can also be written in terms of its components in the x, y, and z directions: \[\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}\]
03

Gravitational Force in the z-direction

In this scenario, the only force acting on the object is the gravitational force in the z-direction, denoted as \(F_g\). The gravitational force can be calculated as: \[F_g = m \cdot g\] where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity, approximately equal to \(9.81\,\text{m/s}^2\). Since the gravitational force only acts in the z-direction, we can write the force vector as: \[\vec{F} = 0\hat{i} + 0\hat{j} + (-F_g)\hat{k}\] Note that we've used a negative sign because the gravitational force acts downward, in the negative z-direction.
04

Apply Newton's Second Law with Gravitational Force

Now we can apply Newton's Second Law to the scenario, with the force vector and the acceleration vector: \[(0\hat{i} + 0\hat{j} + (-F_g)\hat{k}) = m(a_x\hat{i} + a_y\hat{j} + a_z\hat{k})\] By comparing the corresponding components, we get the following equations for the three-dimensional motion: In the x-direction: \[0 = m a_x\Rightarrow a_x=0\] In the y-direction: \[0 = m a_y\Rightarrow a_y=0\] In the z-direction: \[-F_g = m a_z \Rightarrow a_z = -\dfrac{F_g}{m}\]
05

Write the Final Expression

Now we have the expression in the z-direction, replacing \(F_g\) with \(m \cdot g\), we get: \[a_z = -\dfrac{m\cdot g}{m}\] The mass \(m\) cancels out, leaving the final expression for the z-direction acceleration: \[a_z = -g\] So the final expression for Newton's Second Law of Motion in three-dimensional motion with only the gravitational force acting in the z-direction is: \[\vec{a} = 0\hat{i} + 0\hat{j} + (-g)\hat{k}\]

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Most popular questions from this chapter

An object moves on the helix \(\langle\cos t, \sin t, t\rangle,\) for \(t \geq 0\) a. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with a constant speed of \(10 .\) b. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with speed \(t\)

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