Dynamics; Newton’s Laws

of Motion

Force A force is any kind of push or pull on an object. An object at rest needs a force to get it moving; a moving object needs a force to change its velocity.

The magnitude of a force can be measured using a spring scale.

Newton’s First Law of Motion What is the relationship between force and motion?

Every object continues in its state of rest, or of uniform velocity in a straight line, as long as no net force acts on it. The tendency of an object to maintain its state of rest or of uniform velocity in a straight line is called inertia. As a result, Newton's first law is often called the law of inertia.

Inertial Reference Frame

Newton's first law holds ONLY in an inertial reference frame. For most purposes, we usually make the approximation that a reference frame fixed on the Earth is an inertial frame. This excludes rotating and accelerating frames. Any reference frame that moves with constant velocity (say, a car or an airplane) relative to an inertial frame is also an inertial reference frame. Reference frames where the law of inertia does not hold, such as the accelerating reference frames, are called non-inertial reference frames.

Mass Newton used the term mass as a synonym for “quantity of matter.” More precisely, We can say that mass is the measure of inertia of an object. In the SI system, the unit of mass is measured in kilograms (kg).

Mass & Weight

Mass is a property of an object. Weight is the force exerted on that object by gravity. If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same.

NIST (National Institute of Standard and Technology

Newton’s Second Law of Motion What happens if a net force is exerted on an object?

The acceleration of an object is directly proportional to the net force acting on it, and is inversely proportional to the object’s mass. The direction of the acceleration is in the direction of the net force acting on the object.

!a =!F!

m

Force F is a vector, with magnitude and direction.

Newton’s Second Law of Motion Newton's second law relates the description of motion to the force (cause of motion). Force is a vector, so !F = ma is true along each coordinate axis.

!a =!F!

m ------->

Fx = max!Fy = may!Fz = maz!

"

#$$

%$$

The unit of force in the SI system is the newton (N). Note that the pound is a unit of force, not of mass, and can therefore be equated to newtons but not to kilograms. One newton is the force required to impart an acceleration of 1 m/s2 to a mass of 1 kg. Thus 1 N = 1 kg ! m/s2

Example (1)

A book is lying at rest on a table. The book will remain there at rest because:

a)! there is a net force but the book has too much inertia b)! there are no forces acting on it at all c)! It does move, but too slowly to be seen d)! there is no net force on the book e)! there is a net force, but the book is too heavy to move

Example (2)

A hockey puck slides on ice at constant velocity. What is the net force acting on the puck?

a)! More than its weight b)! Equal to its weight c)! Less than its weight but more than zero d)! Depends on the speed of the puck e)! Zero

Example (3)

You put your book on the bus seat next to you. When the bus stops suddenly, the book slides forward off the seat. Why?

a)! A net force acted on it b)! No net force acted on it c)! It remained at rest d)! It did not move, but only seemed to e)! Gravity briefly stopped acting on it

Example (4)

Estimate the net force needed to accelerate (a)! a 1000-kg car at 1/2g

(b)! a 200-gram apple at the same rate.

Example (5)

What average net force is required to bring a 1500-kg car to rest from a speed of 100 km/h within a distance of 55 m?

Example (6)

From rest, we step on the gas of our Ferrari, providing a force F for 4 seconds, speeding it up to a final speed v. If the applied force were only 1/2 F, how long would it have to be applied to reach the same final speed?

a)! 16 s b)! 8 s c)! 4 s d)! 2 s e)! 1 s

v

F

Example (7) From rest, we step on the gas of our Ferrari, providing a force F for 4 seconds. During this time, the car moves 50 m. If the same force would be applied for 8 seconds, how much would the car have traveled during this time?

a)! 250 m b)! 200 m c)! 150 m d)! 100 m e)! 50 m

v

F

Newton’s Third Law of Motion

Where do forces come from? Any time a force is exerted on an object, that force is caused by another object.

Whenever one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first. This law is sometimes paraphrased as “to every action there is an equal and opposite reaction.”

It is very important to remember that the “action” force and the “reaction” force are acting on different objects.

If your hand pushes against the edge of a desk (the force vector is shown in red), the desk pushes back against your hand (the force vector is shown in violet)

Newton’s Third Law of Motion

A key to the correct application of the third law is that the forces are exerted on different objects.

when an ice skater pushes against the wall, the wall pushes back and this force causes her to accelerate away.

We can walk forward because, when one foot pushes backward against the ground, the ground pushes forward on that foot (Newton's third law). The two forces shown act on different objects.

!FGP = !

!FPG

Weight—the Force of Gravity; and the Normal Force

Weight is the force exerted on an object by gravity. Close to the surface of the Earth, where the gravitational force is nearly constant, the weight is:

!FG = m!g

An object at rest must have no net force on it. If it is sitting on a table, the force of gravity is still there; what other force is there?

The force exerted perpendicular to a surface is called the normal force. It is exactly as large as needed to balance the force from the object (if the required force gets too big, something breaks!)

Example (8) A friend has given you a special gift, a box of mass 10.0 kg. The box is resting on the smooth (frictionless) horizontal surface of a table.

(a) Determine the weight of the box and the normal force exerted on it by the table.

(b) Now your friend pushes down on the box with a force of 40.0 N. Again determine the normal force exerted on the box by the table.

(c) If your friend pulls upward on the box with a force of 40.0 N, what now is the normal force exerted on the box by the table?

Example (9) What happens when a person pulls upward on the box with a force equal to, or greater than, the box's weight? For example, let FP = 100.0 N rather than 40.0 N in Example (8).

Example (10)

A 65-kg woman descends in an elevator that briefly accelerates at 0.20g downward. She stands on a scale that reads in kg. During this acceleration, what is her weight and what does the scale read?

Solving Problems with Newton’s Laws -- Free-Body Diagrams

①! Draw a sketch.

②! For one object, draw a free-body diagram, showing all the forces acting on the object. Make the magnitudes and directions as accurate as you can. Label each force. If there are multiple objects, draw a separate diagram for each.

③! Resolve vectors into components.

④! Apply Newton’s second law to each component.

⑤! Solve.

Calculate the sum of the two forces exerted on the boat by workers A & B.

Example (11)

Solving Problems: Tension in a Flexible Cord

When a cord or rope pulls on an object, it is said to be under tension, and the force it exerts is called a tension force.

FxB = FT = mBaB!FxA = Fp ! FT = mAaA"

mA +mB( )a = FP ! FT + FT = FP

Example (12) A system of two objects suspended over a pulley by a flexible cable is referred to as an Atwood machine. An elevator is the real-life application of such system. For the elevator system shown in the picture, find (a)! the acceleration of the elevator (b)! the tension in the cable.

Problems Involving Friction, Inclines

On a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way.

For kinetic—sliding—friction, we write: µk is the coefficient of kinetic friction, and is different for every pair of surfaces.

Ffr = µkFN

Static friction is the frictional force between two surfaces that are not moving along each other. Static friction keeps objects on inclines from sliding, and keeps objects from moving when a force is first applied.

Ffr ! µsFN

Problems Involving Friction, Inclines

An object sliding down an incline has three forces acting on it: the normal force, gravity, and the frictional force.

!! The normal force is always perpendicular to the surface.

!! The friction force is parallel to it.

!! The gravitational force points down.

If the object is at rest, the forces are the same except that we use the static frictional force, and the sum of the forces is zero.

Example (13) An object has begun descending the 30° slope. If the coefficient of kinetic friction is 0.10, what is the objects acceleration?