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Chapter 5: Problem 4

While two forces act on it, a particle is to move at the constant velocity \(\vec{v}=(3 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}-(4 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\). One of the forces is \(\vec{F}_{1}=(2 \mathrm{~N}) \hat{\mathrm{i}}+\) \((-6 \mathrm{~N}) \hat{\mathrm{j}} .\) What is the other force?

Short Answer

Expert verified
\( \vec{F}_2 = (-2 \mathrm{~N}) \hat{\mathrm{i}} + (6 \mathrm{~N}) \hat{\mathrm{j}} \) to keep constant velocity.

Step by step solution

01

Understand the Problem

The problem involves a particle under two forces moving at a constant velocity. A constant velocity means there is no net force acting on the particle. Therefore, the sum of forces should be zero.
02

Use Newton's First Law

According to Newton's First Law, a body at a constant velocity means the net force is zero. Hence, \( \vec{F}_1 + \vec{F}_2 = \vec{0} \). We need to find \( \vec{F}_2 \).
03

Set Up the Vector Equation

Given \( \vec{F}_1 = (2 \mathrm{~N}) \hat{\mathrm{i}} + (-6 \mathrm{~N}) \hat{\mathrm{j}} \) and \( \vec{F}_1 + \vec{F}_2 = 0 \), we rearrange to find \( \vec{F}_2 = -\vec{F}_1 \).
04

Calculate the Components of Force \( \vec{F}_2 \)

From \( \vec{F}_2 = -\vec{F}_1 \), calculate each component: \( F_{2i} = -2 \mathrm{~N} \) (i-component) and \( F_{2j} = 6 \mathrm{~N} \) (j-component).
05

Write the Expression for \( \vec{F}_2 \)

Putting the components together, \( \vec{F}_2 = (-2 \mathrm{~N}) \hat{\mathrm{i}} + (6 \mathrm{~N}) \hat{\mathrm{j}} \). Thus, \( \vec{F}_2 \) complements \( \vec{F}_1 \) to ensure the net force is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Addition
When dealing with forces, particularly in physics, it's crucial to understand how vector addition works. Vectors are quantities that include both magnitude and direction. Dealing with vectors, whether they represent forces or velocities, involves adding and subtracting them just like numbers, but factoring in their directions is also important.

For example, when two forces act on an object, the total or resultant force is found by adding the individual force vectors together using vector addition. To add two vectors, like the forces \( \vec{F}_1 \) and \( \vec{F}_2 \), you add their corresponding components:
  • i-components are added together
  • j-components are added together
For instance, if \( \vec{F}_1 \) has components \( (2 \text{ N}) \hat{\text{i}}, (-6 \text{ N}) \hat{\text{j}} \), and adding the unknown force \( \vec{F}_2 \) results in a net force of zero, it means combining these components using vector addition shows how forces can balance each other out.
Net Force
Net force is a concept that comes straight from Newton's First Law of Motion. This law tells us that an object will not change its motion unless a net force acts upon it. If something is at a constant velocity, like in the given problem, the net force is zero.

To understand this practically, imagine you are pushing a box across a smooth floor. The amount you push is countered by friction or other forces exactly, causing the box to move steadily without accelerating. This is the essence of a net force of zero. Mathematically, for the forces \( \vec{F}_1 \) and \( \vec{F}_2 \), this is represented as:
  • \( \vec{F}_1 + \vec{F}_2 = \vec{0} \)
This equation implies that whatever force is applied by \( \vec{F}_1 \), \( \vec{F}_2 \) must be equal and opposite to maintain zero net force and allow for constant velocity.
Constant Velocity
What does constant velocity really mean? Simply put, a constant velocity indicates that the speed and direction of an object鈥檚 motion remain unchanged over time. When an object moves with constant velocity, it suggests two essential things:

  • The object is moving with a steady speed, not speeding up or slowing down.
  • In terms of forces, they are perfectly balanced, meaning the net force is zero.
Therefore, in our problem, the particle with a constant velocity of \(3 \text{ m/s} \hat{\text{i}} - 4 \text{ m/s} \hat{\text{j}}\), is experiencing forces that completely cancel each other out.

Constancy in velocity doesn't just happen on its own. It must be sustained by a delicate equilibrium of forces acting on the object, confirming the idea that the object's velocity stays true to its course without deviation.

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Most popular questions from this chapter

An elevator cab and its load have a combined mass of \(1600 \mathrm{~kg}\). Find the tension in the supporting cable when the cab, originally moving downward at \(12 \mathrm{~m} / \mathrm{s}\), is brought to rest with constant acceleration in a distance of \(42 \mathrm{~m}\).

A \(0.340 \mathrm{~kg}\) particle moves in an \(x y\) plane according to \(x(t)=-15.00+2.00 t-4.00 t^{3}\) and \(y(t)=25.00+7.00 t-9.00 t^{2}\), with \(x\) and \(y\) in meters and \(t\) in seconds. At \(t=0.700 \mathrm{~s}\), what are (a) the magnitude and (b) the angle (relative to the positive direction of the \(x\) axis) of the net force on the particle, and (c) what is the angle of the particle's direction of travel?

In In April 1974, John Massis of Belgium managed to move two passenger railroad cars. He did so by clamping his teeth down on a bit that was attached to the cars with a rope and then leaning backward while pressing his feet against the railway ties. The cars together weighed \(700 \mathrm{kN}\) (about 80 tons). Assume that he pulled with a constant force that was \(2.5\) times his body weight, at an upward angle \(\theta\) of \(30^{\circ}\) from the horizontal. His mass was \(80 \mathrm{~kg}\), and he moved the cars by \(1.0 \mathrm{~m}\). Neglecting any retarding force from the wheel rotation, find the speed of the cars at the end of the pull.

The high-speed winds around a tornado can drive projectiles into trees, building walls, and even metal traffic signs. In a laboratory simulation, a standard wood toothpick was shot by pneumatic gun into an oak branch. The toothpick's mass was \(0.13 \mathrm{~g}\), its speed before entering the branch was \(220 \mathrm{~m} / \mathrm{s}\), and its penetration depth was \(15 \mathrm{~mm}\). If its speed was decreased at a uniform rate, what was the magnitude of the force of the branch on the toothpick?

Three forces act on a particle that moves with unchanging velocity \(\vec{v}=(2 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}-(7 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\). Two of the forces are \(\vec{F}_{1}=(2 \mathrm{~N}) \hat{\mathrm{i}}+\) \((3 \mathrm{~N}) \hat{\mathrm{j}}+(-2 \mathrm{~N}) \hat{\mathrm{k}}\) and \(\vec{F}_{2}=(-5 \mathrm{~N}) \hat{\mathrm{i}}+(8 \mathrm{~N}) \hat{\mathrm{j}}+(-2 \mathrm{~N}) \hat{\mathrm{k}}\). What is the third force?
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