Learning Goals for Chapter 4 Looking forward at … what the concept of force means in physics, why forces are vectors, and the significance of the net force on an object. what happens when the net force on an object is zero, and the significance of inertial frames of reference. how the acceleration of an object is determined by the net force on the object and the object’s mass. the difference between the mass of an object and its weight. how the forces that two objects exert on each other are related.

Learning Goals for Chapter 5 Looking forward at … how to use Newton’s first law to solve problems involving the forces that act on a body in equilibrium. how to use Newton’s second law to solve problems involving the forces that act on an accelerating body. the nature of the different types of friction forces and how to solve problems that involve these forces. how to solve problems involving the forces that act on a body moving along a circular path. the key properties of the four fundamental forces of nature.

What are some properties of a force?

There are four common types of forces: Normal The normal force is a contact force.

There are four common types of forces: Friction Friction is a contact force.

There are four common types of forces: Tension Tension is a contact force.

There are four common types of forces: Weight Weight is a long-range force.

Drawing force vectors The figure shows a spring balance being used to measure a pull that we apply to a box. We draw a vector to represent the applied force. The length of the vector shows the magnitude; the longer the vector, the greater the force magnitude.

Superposition of forces Several forces acting at a point on an object have the same effect as their vector sum acting at the same point.

Decomposing a force into its component vectors Choose perpendicular x- and y-axes. Fx and Fy are the components of a force along these axes. Use trigonometry to find these force components.

Notation for the vector sum The vector sum of all the forces on an object is called the resultant of the forces or the net force:

Newton’s first law When a body is either at rest or moving with constant velocity (in a straight line with constant speed), we say that the body is in equilibrium. For a body to be in equilibrium, it must be acted on by no forces, or by several forces such that their vector sum—that is, the net force—is zero:

When is Newton’s first law valid? Suppose you are in a bus that is traveling on a straight road and speeding up. If you could stand in the aisle on roller skates, you would start moving backward relative to the bus as the bus gains speed. It looks as though Newton’s first law is not obeyed; there is no net force acting on you, yet your velocity changes. The bus is accelerating with respect to the earth and is not a suitable frame of reference for Newton’s first law. A frame of reference in which Newton’s first law is valid is called an inertial frame of reference.

Using Newton’s first law when forces are in equilibrium A body is in equilibrium when it is at rest or moving with constant velocity in an inertial frame of reference. The essential physical principle is Newton’s first law:

Problem-solving strategy for equilibrium situations Identify the relevant concept: You must use Newton’s first law. Set up the problem by using the following steps: Draw a sketch of the physical situation. Draw a free-body diagram for each body that is in equilibrium. Ask yourself what is interacting with the body by contact or in any other way. If the mass is given, use w = mg to find the weight. Check that you have only included forces that act on the body. Choose a set of coordinate axes and include them in your free-body diagram.

Free-body diagrams

Problem-solving strategy Execute the solution as follows: Find the components of each force along each of the body’s coordinate axes. Set the sum of all x-components of force equal to zero. In a separate equation, set the sum of all y-components equal to zero. If there are two or more bodies, repeat all of the above steps for each body. If the bodies interact with each other, use Newton’s third law to relate the forces they exert on each other. Make sure that you have as many independent equations as the number of unknown quantities. Then solve these equations to obtain the target variables. Evaluate your answer.

An object undergoing uniform circular motion As we have already seen, an object in uniform circular motion is accelerated toward the center of the circle. So the net force on the object must point toward the center of the circle.

Newton’s second law of motion The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object. The SI unit for force is the newton (N). 1 N = 1 kg·m/s2

Systems of units: Table 4.2 We will use the SI system. In the British system, force is measured in pounds, distance in feet, and mass in slugs. In the cgs system, mass is in grams, distance in centimeters, and force in dynes.

Force and acceleration The acceleration of an object is directly proportional to the net force on the object.

Force and acceleration The acceleration of an object is directly proportional to the net force on the object.

Mass and acceleration The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed.

Mass and acceleration The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed.

Mass and acceleration The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed.

Mass and weight The weight of an object (on the earth) is the gravitational force that the earth exerts on it. The weight w of an object of mass m is: The value of g depends on altitude. On other planets, g will have an entirely different value than on the earth.

Relating the mass and weight of a body

Using Newton’s second law: dynamics of particles In dynamics problems, we apply Newton’s second law to bodies on which the net force is not zero. These bodies are not in equilibrium and hence are accelerating:

A note on free-body diagrams does not belong in a free-body diagram.

A note on free-body diagrams Correct free-body diagram

A note on free-body diagrams Incorrect free-body diagram

Problem-solving strategy for dynamics situations Identify the relevant concept: You must use Newton’s second law. Set up the problem by using the following steps: Draw a simple sketch of the situation that shows each moving body. For each body, draw a free-body diagram that shows all the forces acting on the body. Label each force. Usually, one of the forces will be the body’s weight w = mg. Choose your x- and y-coordinate axes for each body, and show them in your free-body diagram. Identify any other equations you might need. If more than one body is involved, there may be relationships among their motions; for example, they may be connected by a rope.

Problem-solving strategy for dynamics situations Execute the solution as follows: For each body, determine the components of the forces along each of the body’s coordinate axes. List all of the known and unknown quantities. In your list, identify the target variable or variables. For each body, write a separate equation for each component of Newton’s second law. Write any additional equations that you identified in step 4 of “Set Up.” (You need as many equations as there are target variables.) Do the easy part—the math! Solve the equations to find the target variable(s). Evaluate your answer.

Apparent weight and apparent weightlessness When a passenger with mass m rides in an elevator with y-acceleration ay, a scale shows the passenger’s apparent weight to be: n = m(g + ay) The extreme case occurs when the elevator has a downward acceleration ay = −g — that is, when it is in free fall. In that case n = 0 and the passenger seems to be weightless. Similarly, an astronaut orbiting the earth with a spacecraft experiences apparent weightlessness.

Frictional forces When a body rests or slides on a surface, the friction force is parallel to the surface.

Frictional forces Friction between two surfaces arises from interactions between molecules on the surfaces.

Kinetic and static friction Kinetic friction acts when a body slides over a surface. The kinetic friction force is fk = µkn. Static friction acts when there is no relative motion between bodies. The static friction force can vary between zero and its maximum value: fs ≤ µsn.

Static friction followed by kinetic friction Before the box slides, static friction acts. But once it starts to slide, kinetic friction acts.

Static friction followed by kinetic friction Before the box slides, static friction acts. But once it starts to slide, kinetic friction acts.

Some approximate coefficients of friction

Newton’s third law

A paradox? If an object pulls back on you just as hard as you pull on it, how can it ever accelerate?

Newton’s third law The simple act of walking depends crucially on Newton’s third law. To start moving forward, you push backward on the ground with your foot. As a reaction, the ground pushes forward on your foot (and hence on your body as a whole) with a force of the same magnitude. This external force provided by the ground is what accelerates your body forward.

Dynamics of circular motion If a particle is in uniform circular motion, both its acceleration and the net force on it are directed toward the center of the circle. The net force on the particle is Fnet = mv2/R.

What if the string breaks? If the string breaks, no net force acts on the ball, so it obeys Newton’s first law and moves in a straight line.

Avoid using “centrifugal force” Figure (a) shows the correct free-body diagram for a body in uniform circular motion. Figure (b) shows a common error. In an inertial frame of reference, there is no such thing as “centrifugal force.”

A car rounds a banked curve At what angle should a curve be banked so a car can make the turn even with no friction? Follow Example 5.22.

The fundamental forces of nature According to current understanding, all forces are expressions of four distinct fundamental forces: gravitational interactions electromagnetic interactions the strong interaction the weak interaction Physicists have taken steps to unify all interactions into a theory of everything.