Lecture 4 Dynamics: Newtons Laws of Motion Objects undergo motion and accelerations. This is caused by an interaction between bodies. Such interactions are called forces. Recall most of our forces are contact forces Newton’s laws of motion: Published Newton’s Principia (1642) SHOW DEMO ILLUSTRATING INERTIA AND FORCE

Examples of Contact Forces: A push or pull can be a force Normal force Tension in a string Friction Two balls colliding Example of Noncontact forces Gravitational force Electric force

NEWTON’S FIRST LAW An object continues in a state of rest or of motion at constant speed in a straight line unless acted upon by a net force. If you push it and let it go, it will move at constant velocity. Show some demos Demo: Air levitated hockey puck sliding across table top Demo: Heavy kilogram mass on table top Example : Hockey puck of mass m on ice Block of Ice

Newton’s First Law of Motion Conceptual Example : Newton’s first law. A school bus comes to a sudden stop, and all of the backpacks on the floor start to slide forward. What force causes them to do that? Answer: No force; the backpacks continue moving until stopped by friction or collision.

NEWTON’S FIRST LAW An object continues in a state of rest or of motion at constant speed in a straight line unless acted upon by a net force. If you don’t push it, it won’t move. A body has inertia. Show some demos Demo: Air levitated hockey puck sliding across table top Demo: Heavy kilogram mass on table top Example : Hockey puck of mass m on ice Block of Ice

Inertia Another way of understanding the First Law Inertia is a bodies resistance to change due to forces. Inertia is related to mass. Some examples of inertia Hit a nail in a piece of wood on an anvil sitting on your head Mass on string (Demo)

Two different ways of breaking the string: Inertia and Tension Upper string breaks when you pull slowly because tension is greater Lower string breaks when you pull quickly because of inertia First pull fast see where it breaks Then pull slowly see where it breaks Explain

NEWTON’S SECOND LAW Sum of all external forces acting on the body = net force on body of mass m Newton’s second law is much more general than the first. It tells us what happens when forces acting on a body are present. A net force acting on a body from the environment produces an acceleration of the body. The inertia of the body, as measured by its mass, resists the change in velocity caused by the force. The greater the mass, the smaller the acceleration. It is simplest to think about Newton’s laws in the absence of friction. We can take friction into account, however, by including it as one of the forces acting on the body of interest. SHOW Propeller driven glider timed for one unit of distance and four units. Since it takes twice as long to go four times as far, the acceleration must be constant. So the propeller is providing a nearly constant force, leading to a constant acceleration. System Mass Acceleration Force SI kg m/s2 Newton (N) CGS g cm/s2 dyne (dyn) BE slug (sl) ft/s2 pound (lb)

Because force is a vector we have 3 equations relating force and acceleration each corresponding To a different coordinate

Newton’s Second Law of Motion Example 4-2: Force to accelerate a fast car. Estimate the net force needed to accelerate (a) a 1000-kg car at ½ g Figure 4-6. 4-2. Use Newton’s second law: acceleration is about 5 m/s2, so F is about 5000 N for the car and 1 N for the apple. 4-3. First, find the acceleration (assumed constant) from the initial and final speeds and the stopping distance; a = -7.1 m/s2. Then use Newton’s second law: F = -1.1 x 104 N.

Example Force to stop a car. What average net force is required to bring a 1500-kg car to rest from a speed of 30 m/s within a distance of 50 m?

NEWTON’S THIRD LAW When two bodies interact, the forces on the bodies due to each other are always equal in magnitude and opposite in direction. M Table

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. Make sure you don’t use them as if they were acting on the same object. Figure 4-9. Caption: An example of Newton’s third law: when an ice skater pushes against the wall, the wall pushes back and this force causes her to accelerate away.

Newton’s Third Law of Motion Helpful notation: the first subscript is the object that the force is being exerted on; the second is the source. Figure 4-11. Caption: 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.

Rules for drawing free body diagrams Rules for drawing free body diagrams. The purpose is to Isolate the forces acting on one body Draw a diagram Represent the body by a point. Each force acting on the body is represented by a vector with tail at the point and the length of vector indicating the approximate magnitude of the force. It may be convenient to resolve the forces into components Apply Newton’s second law and solve for the unknowns

Weight and the force of gravity Falling objects accelerate at the rate of 9.81 m/s2 or g. From Newtons 2nd law, we know F= ma where a =g. If a table or floor is in the way gravity is still acting and trying to accelerate the object. This produces the gravitational force acting on the object called weight. But the object is not accelerating. Isn’t this a violation of Newtons 2nd law NO. Because the table or floor exerts an upward force back on the object so that the net force is 0. This upward force is called the Normal force.

What is the free body diagram of a block at rest on the table? When you stand still on a sidewalk, you are not accelerating along the sidewalk, so there is no force on you in that direction. In that case the only force the sidewalk exerts on you is upward, opposing gravity. This force is perpendicular to the sidewalk surface. This is called a normal force. Normal is a term that refers to something being perpendicular to a surface or line. W N

Book leaning against a crate on a table at rest Book leaning against a crate on a table at rest. What are the action –reaction pairs because of Newtons 3rd Law? Table T

Draw a free body diagram of the forces acting on the crate NT mg B 2) Does the crate C accelerate?

A crate of mass 310 kg is being pulled by a man as shown in the figure A crate of mass 310 kg is being pulled by a man as shown in the figure. What is the acceleration of the crate along the x direction? Man does not move. N T f +y N +x W First draw a free body diagram of the forces acting on the crate

Draw a free body diagram of the forces acting on the crate y is vertical x is horizontal N T f W Now we want to apply Newton’s second Law to the x and y components independently.

A crate of mass 310 kg is being pulled by a man as shown in the figure A crate of mass 310 kg is being pulled by a man as shown in the figure. What is the acceleration of the crate along the x direction? Man does not move. +y N W T f N +x x component of forces in free body diagram

What is the normal force assuming there is no acceleration in the y direction? N W T f N y component of forces in free body diagram

Problem: What is the acceleration of the system of the two blocks and the contact force between the blocks? What is the net force on Block B? 31 kg 65 N 24 kg A B B A 65 N 24

Problem: What is the acceleration of the system of the two blocks and the contact force between the blocks? What is the net force on Block B? 31 kg 65 N 24 kg A B 65 N A B Student Version 25

Now lets look at tension in a string Tension in the string is equal to the weight = 10 N The scale reads the tension in the string

Is the tension in the string any different when I have weights pulling it down on both sides?

Problem 12 kg 24 kg 31 kg T3=65 N kg 31 kg 12 kg 24 kg 65 N What is the acceleration of the system? Find T1 Find T2 a) b) c)

Rev George Atwood’s machine 1746 -1807 Tutor Trinity College, Cambridge 2 DEMO Atwoods Machine

Set up Coordinate system Assume acceleration in some direction Draw free body diagram for each body Apply Newtons second law to each body mg Mg T a 1) Assume mass 1 or M is going down 1 2 +y +x DEMO Atwoods Machine

Free body diagram for each body and apply Newtons second law 1. 2. Free body diagram for each body and apply Newtons second law mg Mg T a 1 2 +y +x DEMO Atwoods Machine Solve eq 1 and 2 for T and a

Free body diagram for each body and apply Newtons second law 1. 2. Free body diagram for each body and apply Newtons second law mg Mg T a 1 2 +y +x DEMO Atwoods Machine

4-7 Solving Problems with Newton’s Laws: Free-Body Diagrams Example 4-13: Elevator and counterweight (Atwood’s machine). A system of two objects suspended over a pulley by a flexible cable is sometimes referred to as an Atwood’s machine. Here, let the mass of the counterweight be 1000 kg. Assume the mass of the empty elevator is 850 kg, and its mass when carrying four passengers is 1150 kg. For the latter case calculate (a) the acceleration of the elevator and (b) the tension in the cable. Figure 4-23. Caption: Example 4–13. (a) Atwood’s machine in the form of an elevator–counterweight system. (b) and (c) Free-body diagrams for the two objects. Answer: Each mass has two forces on it, gravity pulling downward and the tension in the cable pulling upward. The tension in the cable is the same for both, and both masses have the same acceleration. Writing Newton’s second law for each mass gives us two equations; there are two unknowns, the acceleration and the tension. Solving the equations for the unknowns gives a = 0.68 m/s2 and FT = 10,500 N.

Another example. Find tension T in the cord and the downward acceleration a. Draw Free Body Diagram of each body M and m assuming m is accelerating down: Frictionless pulley T Demos for inclined planes, pulleys Normal force and tension Example problems Atwoods machine, Einstein in an elevator or accelerating scale, blocks accelerating T

Set up Coordinate system Assume acceleration in some direction Draw free body diagram for each body Apply Newtons second law to each body Frictionless pulley +y +x Demos for inclined planes, pulleys Normal force and tension Example problems Atwoods machine, Einstein in an elevator or accelerating scale, blocks accelerating T a a

+y +x Demos for inclined planes, pulleys Normal force and tension Example problems Atwoods machine, Einstein in an elevator or accelerating scale, blocks accelerating

Demos for inclined planes, pulleys Normal force and tension Example problems Atwoods machine, Einstein in an elevator or accelerating scale, blocks accelerating

Tug-of-war demo illustrates how a small sideways force can produce a large horizontal force Suppose two guys in the tug of war are at 4.5 meters apart and I pull the rope out 0.15 meters. Then φ =4 degrees 2.25 m f Therefore, The smaller the angle the larger the magnification If then