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Chapter 6 Dynamics I: Motion Along a Line

Chapter Goal: To learn how to solve linear force-and-motion problems.Slide 6-2

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An object on which the net force is zero is in equilibrium.

If the object is at rest, it is in static equilibrium.

If the object is moving along a straight line with a constant velocity it is in dynamic equilibrium.

The requirement for either type of equilibrium is:

Equilibrium

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The concept of equilibrium is essential for the engineering analysis of stationary objects such as bridges.

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The figure shows the view looking down onto a sheet of frictionless ice. A puck, tied with a string to point P, slides on the ice in the circular path shown and has made many revolutions. If the string suddenly breaks with the puck in the position shown, which path best represents the puck’s subsequent motion?

QuickCheck 6.1

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The figure shows the view looking down onto a sheet of frictionless ice. A puck, tied with a string to point P, slides on the ice in the circular path shown and has made many revolutions. If the string suddenly breaks with the puck in the position shown, which path best represents the puck’s subsequent motion?

QuickCheck 6.1

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Newton’s first law!

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A ring, seen from above, is pulled on by three forces. The ring is not moving. How big is the force F?

A. 20 N

B. 10cos N

C. 10sin N

D. 20cos N

E. 20sin N

QuickCheck 6.2

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A ring, seen from above, is pulled on by three forces. The ring is not moving. How big is the force F?

A. 20 N

B. 10cos N

C. 10sin N

D. 20cos N

E. 20sin N

QuickCheck 6.2

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Example 6.2 Towing a Car up a Hill

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Example 6.2 Towing a Car up a Hill

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Example 6.2 Towing a Car up a Hill

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A car is parked on a hill. Which is the correct free-body diagram?

QuickCheck 6.3

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A car is parked on a hill. Which is the correct free-body diagram?

QuickCheck 6.3

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The essence of Newtonian mechanics can be expressed in two steps:

The forces on an object determine its  acceleration , and

The object’s trajectory can be determined by  using in the equations of kinematics.

Using Newton’s Second Law

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Example 6.3 Speed of a Towed Car

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Example 6.3 Speed of a Towed Car

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Example 6.3 Speed of a Towed Car

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Example 6.3 Speed of a Towed Car

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Mass: An Intrinsic Property

A pan balance, shown in the figure, is a device for measuring mass.

The measurement does not depend on the strength of gravity.

Mass is a scalar quantity that describes an object’s inertia.

Mass describes the amount of matter in an object.

Mass is an intrinsic property of an object.Slide 6-54

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Gravity: A Force

Gravity is an attractive, long-range force between any two objects.

The figure shows two objects with masses m1 and m2 whose centers are separated by distance r.

Each object pulls on the other with a force:

where G = 6.67 × 10−11 N m2/kg2 is the gravitational constant.

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Gravity: A Force

The gravitational force between two human-sized objects is very small.

Only when one of the objects is planet-sized or larger does gravity become an important force.

For objects near the surface of the planet earth:

where M and R are the mass and radius of the earth, and g = 9.80 m/s2.

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Gravity: A Force

The magnitude of the gravitational force is FG = mg, where:

All objects on the same planet, regardless of mass, have the same free-fall acceleration!

The figure shows the free-body diagram of an object in free fall near the surface of a planet.

With , Newton’s second law predicts the acceleration to be:

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Weight: A Measurement

You weigh apples in the grocery store by placing them in a spring scale and stretching a spring.

The reading of the spring scale is the magnitude of Fsp.

We define the weight of an object as the reading Fsp of a calibrated spring scale on which the object is stationary.

Because Fsp is a force, weight is measured in newtons.

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Weight: A Measurement

A bathroom scale uses compressed springs which push up.

When any spring scale measures an object at rest, .

The upward spring force exactly balances the downward gravitational force of magnitude mg:

Weight is defined as the magnitude of Fsp when the object is at rest relative to the stationary scale:

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Weight: A Measurement

The figure shows a man weighing himself in an accelerating elevator.

Looking at the free-body diagram, the y-component of Newton’s second law is:

The man’s weight as he accelerates vertically is:

You weigh more as an elevator accelerates upward!

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A. > 490 N.

B. 490 N.

C. < 490 N but not 0 N.

D. 0 N.

QuickCheck 6.8

A 50-kg student (mg = 490 N) gets in a 1000-kg elevator at rest and stands on a metric bathroom scale. As the elevator accelerates upward, the scale reads

Slide 6-63

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A. > 490 N.

B. 490 N.

C. < 490 N but not 0 N.

D. 0 N.

QuickCheck 6.8

A 50-kg student (mg = 490 N) gets in a 1000-kg elevator at rest and stands on a metric bathroom scale. As the elevator accelerates upward, the scale reads

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Weightlessness

The weight of an object which accelerates vertically is

If an object is accelerating downward with ay = –g, then w = 0.

An object in free fall has no weight!

Astronauts while orbiting the earth are also weightless.

Does this mean that they are in free fall?

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Astronauts are weightless as they orbit the earth.

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A. > 490 N.

B. 490 N.

C. < 490 N but not 0 N.

D. 0 N.

QuickCheck 6.10

A 50-kg student (mg = 490 N) gets in a 1000-kg elevator at rest and stands on a metric bathroom scale. Sadly, the elevator cable breaks. What is the student’s weight during the few second it takes the student to plunge to his doom?

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A. > 490 N.

B. 490 N.

C. < 490 N but not 0 N.

D. 0 N.

QuickCheck 6.10

A 50-kg student (mg = 490 N) gets in a 1000-kg elevator at rest and stands on a metric bathroom scale. Sadly, the elevator cable breaks. What is the student’s weight during the few second it takes the student to plunge to his doom?

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The bathroom scale would read 0 N. Weight is reading of a scale on which the object is stationary relative to the scale.

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Static Friction

A shoe pushes on a wooden floor but does not slip.

On a microscopic scale, both surfaces are “rough” and high features on the two surfaces form molecular bonds.

These bonds can produce a force tangent to the surface, called the static friction force.

Static friction is a result of many molecular springs being compressed or stretched ever so slightly.

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Static Friction

The figure shows a person pushing on a box that, due to static friction, isn’t moving.

Looking at the free-body diagram, the x-component of Newton’s first law requires that the static friction force must exactly balance the pushing force:

points in the direction opposite to the way the object would move if there were no static friction.

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Static Friction

Static friction force has a maximum possible size fs max.

An object remains at rest as long as fs < fs max. The object just begins to slip when fs = fs max. A static friction force fs > fs max is not physically

possible.

  where the proportionality constant μs is called the coefficient of static friction.

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Kinetic Friction

The kinetic friction force is proportional to the magnitude of the normal force:

where the proportionality constant μk is called the coefficient of kinetic friction.

The kinetic friction direction is opposite to the velocity of the object relative to the surface.

For any particular pair of surfaces, μk < μs.

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QuickCheck 6.15

A box is being pulled to the right at steady speed by a rope that angles upward. In this situation:

A. n > mg.

B. n = mg.

C. n < mg.

D. n = 0.

E. Not enough information to judge the size of the normal force.

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QuickCheck 6.15

A box is being pulled to the right at steady speed by a rope that angles upward. In this situation:

A. n > mg.

B. n = mg.

C. n < mg.

D. n = 0.E. Not enough information to

judge the size of the normal force.

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Example 6.10 Make Sure the Cargo Doesn’t Slide

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Example 6.10 Make Sure the Cargo Doesn’t Slide

MODEL Let the box, which we’ll model as a particle, be the object of interest. Only the truck exerts contact forces on the box. The box does not slip relative to the truck. If the truck bed were frictionless, the box would slide backward as

seen in the truck’s reference frame as the truck accelerates. The force that prevents sliding is static friction.

The box must accelerate forward with the truck: abox = atruck.

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Example 6.10 Make Sure the Cargo Doesn’t Slide

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Example 6.10 Make Sure the Cargo Doesn’t Slide

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Example 6.10 Make Sure the Cargo Doesn’t Slide

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Example 6.10 Make Sure the Cargo Doesn’t Slide

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