Chapter 6 Dynamics I: Motion Along a Line

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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Equilibrium 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: The concept of equilibrium is essential for the engineering analysis of stationary objects such as bridges. Slide 6-27

Problem-Solving Strategy: Equilibrium Problems Slide 6-30

Problem-Solving Strategy: Equilibrium Problems Slide 6-31

Example 6.2 Towing a Car up a Hill Slide 6-34

Example 6.2 Towing a Car up a Hill Slide 6-35

Example 6.2 Towing a Car up a Hill Slide 6-36

Example 6.2 Towing a Car up a Hill Slide 6-37

Example 6.2 Towing a Car up a Hill Slide 6-38

Using Newton s Second Law 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. Slide 6-43

Problem-Solving Strategy: Dynamics Problems Slide 6-44

Problem-Solving Strategy: Dynamics Problems Slide 6-45

Example 6.3 Speed of a Towed Car Slide 6-48

Example 6.3 Speed of a Towed Car Slide 6-49

Example 6.3 Speed of a Towed Car Slide 6-50

Example 6.3 Speed of a Towed Car Slide 6-51

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

Gravity: A Force Gravity is an attractive, long-range force between any two objects. The figure shows two objects with masses m 1 and m 2 whose centers are separated by distance r. Each object pulls on the other with a force: where G = 6.67 10 11 N m 2 /kg 2 is the gravitational constant. Slide 6-55

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/s 2. Slide 6-56

Gravity: A Force The magnitude of the gravitational force is F G = mg, where: 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: All objects on the same planet, regardless of mass, have the same free-fall acceleration! Slide 6-57

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 F sp. We define the weight of an object as the reading F sp of a calibrated spring scale on which the object is stationary. Because F sp is a force, weight is measured in newtons. Slide 6-58

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 F sp when the object is at rest relative to the stationary scale: Slide 6-59

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! Slide 6-62

Weightlessness The weight of an object which accelerates vertically is If an object is accelerating downward with a y = 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? Astronauts are weightless as they orbit the earth. Slide 6-67

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. Slide 6-72

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. Slide 6-73

Static Friction Static friction acts in response to an applied force. Slide 6-74

Static Friction Static friction force has a maximum possible size f s max. An object remains at rest as long as f s < f s max. The object just begins to slip when f s = f s max. A static friction force f s > f s max is not physically possible. where the proportionality constant μ s is called the coefficient of static friction. Slide 6-77

Kinetic Friction The kinetic friction force is proportional to the magnitude of the normal force: where the proportionality constant μ k 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, μ k < μ s. Slide 6-80

Rolling Motion If you slam on the brakes so hard that the car tires slide against the road surface, this is kinetic friction. Under normal driving conditions, the portion of the rolling wheel that contacts the surface is stationary, not sliding. If your car is accelerating or decelerating or turning, it is the static friction of the road on the wheels that provides the net force which accelerates the car. Slide 6-87

Rolling Friction A car with no engine or brakes applied does not roll forever; it gradually slows down. This is due to rolling friction. The force of rolling friction can be calculated as: where μ r is called the coefficient of rolling friction. The rolling friction direction is opposite to the velocity of the rolling object relative to the surface. Slide 6-88

Coefficients of Friction Slide 6-89

A Model of Friction The actual causes of friction involve microscopic surface properties and molecular bonds. Experiments show that reasonable predictions are produced by a model of friction a simplification of reality: Here motion means motion relative to the surface. Forces of kinetic and rolling friction are proportional to the normal force of the surface on the object. The maximum static friction force is proportional to the normal force of the surface on the object. Slide 6-90

A Model of Friction The friction force response to an increasing applied force. Slide 6-91

Causes of Friction All surfaces are very rough on a microscopic scale. When two surfaces are pressed together, the high points on each side come into contact and form molecular bonds. The amount of contact depends on the normal force n. When the two surfaces are sliding against each other, the bonds don t form fully, but they do tend to slow the motion. Slide 6-92

Drag The air exerts a drag force on objects as they move through the air. Faster objects experience a greater drag force than slower objects. The drag force on a high-speed motorcyclist is significant. The drag force direction is opposite the object s velocity. Slide 6-93

Drag For normal-sized objects on earth traveling at a speed v which is less than a few hundred meters per second, air resistance can be modeled as: A is the cross-section area of the object. ρ is the density of the air, which is about 1.2 kg/m 3. C is the drag coefficient, which is a dimensionless number that depends on the shape of the object. Slide 6-94

Drag Cross-section areas for objects of different shape. Slide 6-95

Example 6.7 Air Resistance Compared to Rolling Friction Slide 6-96

Example 6.7 Air Resistance Compared to Rolling Friction Slide 6-97

Example 6.7 Air Resistance Compared to Rolling Friction Slide 6-98

Example 6.7 Air Resistance Compared to Rolling Friction Slide 6-99

Terminal Speed The drag force from the air increases as an object falls and gains speed. If the object falls far enough, it will eventually reach a speed at which D = F G. At this speed, the net force is zero, so the object falls at a constant speed, called the terminal speed v term. Slide 6-100

Terminal Speed The figure shows the velocity-versus-time graph of a falling object with and without drag. Without drag, the velocity graph is a straight line with a y = g. When drag is included, the vertical component of the velocity asymptotically approaches v term. Slide 6-101

Example 6.10 Make Sure the Cargo Doesn t Slide Slide 6-102

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: a box = a truck. Slide 6-103