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Chapter 4: Problem 18

When you stand still, there are two forces acting on you, the force of gravity (your weight) and the normal force of the floor pushing up on your feet. Are these forces equal and opposite? Does Newton's third law relate them to each other? Explain.

Short Answer

Expert verified
The gravitational and normal forces are equal and opposite but are not action-reaction pairs under Newton's third law.

Step by step solution

01

Understanding Newton's Third Law and Forces

Newton's third law states that for every action, there is an equal and opposite reaction. This means if object A exerts a force on object B, object B simultaneously exerts a force of equal magnitude and opposite direction on object A. We need to understand how this relates to the forces on a person standing still.
02

Identifying the Forces Involved

When you stand still, there are two primary forces acting on you: the gravitational force (weight) pulling you downwards and the normal force from the ground pushing you upwards. The gravitational force is given by your mass times the acceleration due to gravity, \( F_{gravity} = mg \), and the normal force is the force exerted by the ground to support your weight.
03

Determining if the Forces are Equal and Opposite

For a person standing still, the net force must be zero for equilibrium. This means the gravitational force downward is balanced by the normal force upward. Hence, these two forces are equal in magnitude and opposite in direction, fulfilling the condition for equilibrium.
04

Analysing the Relation to Newton's Third Law

Newton's third law specifically deals with forces between two interacting bodies. While the normal force and the gravitational force are equal and opposite, they act on the same body (you). Therefore, they are not an action-reaction pair under Newton's third law. Instead, the action-reaction pair is between you and the Earth (the gravitational force pulling you and you pulling the Earth), or between you and the ground (you pushing down on the ground and the ground pushing up on you).

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Key Concepts

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

Forces and Equilibrium
When you are standing still, it's a great example to understand the concept of equilibrium. Equilibrium is when all the forces acting on an object are balanced, and as a result, the object stays at rest or moves at a constant velocity. In this case, the gravitational force pulling you downwards is perfectly balanced by the normal force pushing upwards. The sum of these forces in the vertical direction is zero, and that's why you don't accelerate up or down.

Think of equilibrium as a perfect tug-of-war where both sides pull with equal force, keeping the rope perfectly still. It's the same for your body. The balance doesn't mean that there are no forces; it means the forces cancel each other out.
  • Equilibrium leads to no net force, resulting in a state of rest or constant motion.
  • It's crucial for understanding many physics problems where forces must balance each other.
Gravitational Force
The gravitational force is something we experience every day. It's the force that Earth exerts on objects, pulling them towards its center. When standing still, your weight is essentially the gravitational force acting on you. Mathematically, it can be expressed as:
  • Gravitational force = mass 脳 acceleration due to gravity
This is represented by the equation: \[ F_{gravity} = mg \] where \( m \) is your mass, and \( g \) is the acceleration due to gravity, approximately 9.8 m/s虏 on Earth.

This force is always directed downward towards the center of the Earth. It's important to remember that gravitational force is always present, and it plays a key role in everything from keeping planets in orbit to determining how much you weigh.
  • Gravitational force acts downward towards the center of Earth.
  • It's calculated using the mass of the object and the gravitational acceleration value.
Normal Force
The normal force acts in a way that's very intuitive once you start noticing it. This force is the support force exerted by a surface that prevents objects from "falling" through it. When you stand on the floor, the normal force is what keeps you from sinking into the ground. It acts perpendicular to the surface you are standing on.
  • The normal force is equal in size and opposite in direction to the gravitational force, when you are at rest.
  • It accounts for you not falling through the floor.
While it closely balances the gravitational force when you are standing still, it's not part of an action-reaction pair as per Newton's third law. Instead, it reacts to the pressure your body exerts on the floor. This interaction helps maintain balance and stability.

Understanding normal force is crucial in analyzing problems where contact with a surface comes into play, such as when driving a car or sitting on a chair.
  • The normal force is always perpendicular to the contact surface.
  • It increases with more weight or steeper angles of surfaces.

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Most popular questions from this chapter

A wagon is being pulled at constant speed up a slope \(\theta\) by a rope that makes an angle \(\phi\) with the vertical. (a) Assuming negligible friction, show that the tension in the rope is given by the equation $$ T=\frac{\sin \theta}{\sin (\theta+\phi)} m g $$ (b) Interpret this equation in the special cases of \(\phi=0\) and \(\phi=180^{\circ}-\theta\).(solution in the pdf version of the book)

A rocket ejects exhaust with an exhaust velocity \(u\). The rate at which the exhaust mass is used (mass per unit time) is \(b\). We assume that the rocket accelerates in a straight line starting from rest, and that no external forces act on it. Let the rocket's initial mass (fuel plus the body and payload) be \(m_{i}\), and \(m_{f}\) be its final mass, after all the fuel is used up. (a) Find the rocket's final velocity, \(v\), in terms of \(u, m_{i}\), and \(m_{f} .\) Neglect the effects of special relativity. (b) A typical exhaust velocity for chemical rocket engines is \(4000 \mathrm{~m} / \mathrm{s}\). Estimate the initial mass of a rocket that could accelerate a one-ton payload to \(10 \%\) of the speed of light, and show that this design won't work. (For the sake of the estimate, ignore the mass of the fuel tanks. The speed is fairly small compared to \(c\), so it's not an unreasonable approximation to ignore relativity.) (answer check available at lightandmatter.com)

A mass \(m\) moving at velocity \(v\) collides with a stationary target having the same mass \(m\). Find the maximum amount of energy that can be released as heat and sound. (answer check available at lightandmatter.com)

A uranium atom deep in the earth spits out an alpha particle. An alpha particle is a fragment of an atom. This alpha particle has initial speed \(v\), and travels a distance \(d\) before stopping in the earth. (a) Find the force, \(F\), from the dirt that stopped the particle, in terms of \(v, d\), and its mass, \(m\). Don't plug in any numbers yet. Assume that the force was constant.(answer check available at lightandmatter.com) (b) Show that your answer has the right units. (c) Discuss how your answer to part a depends on all three variables, and show that it makes sense. That is, for each variable, discuss what would happen to the result if you changed it while keeping the other two variables constant. Would a bigger value give a smaller result, or a bigger result? Once you've figured out this mathematical relationship, show that it makes sense physically. (d) Evaluate your result for \(m=6.7 \times 10^{-27}\) kg, \(v=2.0 \times 10^{4} \mathrm{~km} / \mathrm{s}\), and \(d=0.71 \mathrm{~mm}\).(answer check available at lightandmatter.com)

Find the angle between the following two vectors: $$ \begin{array}{l} \hat{\mathbf{x}}+2 \hat{\mathbf{y}}+3 \hat{\mathbf{z}} \\ 4 \hat{\mathbf{x}}+5 \hat{\mathbf{y}}+6 \hat{\mathbf{z}} \end{array} $$ \mathrm{\\{} ~ h w h i n t ~ \\{ h w h i n t : a n g l e b e t w e e n \\} ( a n s w e r ~ c h e c k ~ a v a i l a b l e ~ a t ~ l i g h t a n d m a t t e r . c o m ) ~
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