Dynamics • Galileo

–An object will stay in motion unless friction is present.–Friction acts between surfaces of objects slowing them down.

• Inertia means “resistance to being accelerated”.

• Forces: –pushes and pulls exerted on objects by other objects.–Measured in pounds (lbs) or “S.I.” units of newtons (N)– 1N is about equal to ¼ pound.

Newton’s First Law of Motion(“Law of Inertia”)

• An object at rest or at some velocity will remain at rest or at that velocity unless acted upon by an external net force.

• An object is in static equilibrium if– It is at rest.– It is acted upon by no net external force.

• Forces are vectors – I can pull 40 newtons east (or west)– we add them the same way we added the

displacement vectors.

External Force vs. Internal Force

• External Forces on an object are forces applied from outside the object.

• Internal forces are applied from within the object.– For example, If I pull on a book, the molecules I’m

touching are pulling in turn on molecules behind them. These “intermolecular pulls” are “internal forces”.

Mass and Weight

• Mass is a measure of the amount of material in an object. (a “chemical” definition.) – S.I. unit is kilograms (kg)

• The weight (w) of an object is the gravitational force (Fg) exerted by the Earth on that object.– Units of weight are newtons (N) (it’s a force)– Near the Earth’s surface: w = Fg = mg,

• where m = object’s mass (kg)• g = 10 (m/s)/s = 10 m/s2

• Please don’t confuse MASS with WEIGHT.– It’s a very common error.

Calculations with Mass and Weight

• What is the weight of a block with a mass of 1 kg?

w = mg = (1 kg) (10 m/s2) = 10 N = 2.2 pounds.

(So: 1 kilogram weighs 10 newtons or 2.2 pounds)• What is the force of gravity on this block due to the Earth?

Fg = w = 10 N = 2.2 pounds.

Remember:

• The force of gravity on an object and the object’s weight are the same thing.

• The mass of an object and the object’s weight are different things.

Which of the following are examples of Static Equilibrium?

• A ball at its highest point after being tossed in the air?– No. The ball is stopped but it has a net force on it

equal to its weight.

• A book at rest on a table?– Yes. The Earth pulls down on it, but the table

pushes back up on it. The net force on it is zero.

• A pendulum at the top of its swing?– No. It has a nonzero net force due to its weight

and the tension in the string.

What is the Net Force on These Objects?

North

East1.

5N10 N

3N

5N

8N

10 N

FNET = 5N East + 5N South

FNET = 5N West

An Object Held up by a String(example 1)

1. How many Forces are acting on the object?2. What is the Net Force on the object?3. What is the Force due to gravity on the object?4. What is the Tension in the string?5. Let’s go through the steps needed to solve this

problem.

The object has a mass (m)

Suppose m = 1 kg.

Free Body Diagram

• The Free Body Diagram is a drawing to help us visualize all of the forces acting on an object.

• You will use the Free Body Diagram throughout your career in physics.

• To draw an FBD:– Draw the object as a point– Draw each force on the object as a vector starting from

the object.– Example:

T

Fg = mg

Steps for solving a problem.

1. Draw the FBD

2. Sum up the forces as vectors to find FNET.

3. If the object is in Equilibrium, then set FNET equal to zero (Applying Newton’s First Law).

4. Solve for the unknown (for example, the tension in the string.)

Let’s apply these steps to the above problem.

Another Problem (example 2)

• Suppose the same mass (m=1kg) is suspended by two strings. What is the tension in each string?

Apply the steps from the last slide.

Another Problem(example 3)

• Suppose the object is suspended by two strings, but they are each at an angle

• Does the tension change? If so, how?

How do we find the tension in the strings?

(Hint: Use a graphical method)

Newton’s Second Law

• Suppose FNET is not zero. What will happen to the object?• The object will accelerate. • Which will accelerate more under the same FNET :

– A heavy object or a light object?• A light object.• Newton’s 2nd Law of Motion: Net Force = mass * acceleration: FNET = ma• Can this be a new definition of mass?• Mass is a measure of an object’s

– resistance to being accelerated or– “inertia”

• We have just defined “inertial mass”.

Friction

• Force acting between two surfaces in contact.• Static Friction: Object on surface is not moving.

– Friction acts in the direction opposite to the sum of the other forces applied parallel to the surface.

• Kinetic Friction: Object is in motion– Friction acts in the direction opposite to the motion.– Friction will always act so as to slow down the object.

• Air Resistance: – Friction between the surface of an object and the air as

the object moves.– Air resistance increases with increasing speed.

Example

• A crate with a mass of 2 kg is pushed along a surface with a force of 10N to the east. The surface exerts a frictional force of 4N on the object.– What is the net force on the crate?– What is the acceleration of the crate?– How fast is the crate moving after 3 seconds?– What distance has the crate been pushed after 3

seconds?

• Always start with a Free Body Diagram.

Recipe for Solving Problems using Newton’s Second Law

1. Identify known and unknown quantities in the problem.

2. Draw a free body diagram for each object in the problem.

3. Using the FBD, write an expression for the net force on each object.

4. Set the expression for the net force on each object equal to that object’s mass times its acceleration: (FNET = ma)

5. Solve for unknowns in the problem.

A More Complicated Example

• A cart of mass (m1) is on a table attached to one end of a string running over a pulley. The other end of the string is attached to a mass (m2). which hangs out beyond the table top. (Neglect friction)

a. What is the acceleration of the cart (m1)?

b. What is the acceleration of the mass (m2)?c. What is the tension (T) in the string?

(Assume we know m1 and m2; we can put numbers in later.)

• What’s the first thing we do to solve this problem?

• We must make a Free Body Diagram for EACH OBJECT in the system.

Pressure• Pressure = Force applied over an area divided by

the area: P = F/A• Example:

– Standing on your feet:• What is the force of gravity acting on you?• How would you calculate the pressure exerted on the floor?

– Standing on your toes:• What is the force of gravity acting on you?• How would you calculate the pressure exerted on the floor?

– Does standing on your feet or on your toes feel any different?

– Pressure is a measure of the intensity of a force.

Why do All Objects Fall at the Same Rate?

• What is the net force on a free falling object?FNET = mg

• What do we get if we apply Newton’s Second Law to such an object?FNET = ma

• So: mg = ma

• And: g = a

• So the object falls with an acceleration equal to 10 m/s2 regardless of its mass.

Air Resistance and Terminal Velocity

• As an object falls – What happens to the upward force of air resistance?

• Increases because speed increases.

– What happens to the force due to gravity?• Remains the same

– What happens when the force due to air resistance equals the force due to gravity?

• FNET = 0; Object falls at constant (terminal) velocity.

– If you are falling, how can you change your terminal velocity?

• Expose more area downward to increase air resistance; reduce terminal velocity.

• Expose less area downward to decrease air resistance; increase terminal velocity.

Newton’s Third Law• Forces on objects are exerted by other objects.• “Whenever an object exerts a force on a second

object, the second object exerts an equal and opposite force back on the first object.”– We call these forces “action-reaction pairs”.

• Question: Can these forces cancel out?– NO. They are acting on different objects.– Forces can only cancel when acting on the same object.

• Can a Free Body Diagram include both forces in an action-reaction pair?– NO. A free body diagram is made for one object and

includes only the forces acting on that object alone.

The “Normal Force”• When an object is pressed against a surface, the surface

pushes back on the object with a force that is perpendicular (or “normal”) to the surface. We call this force the “normal force”.

Book resting on a table; the table pushes upward, normal to the table’s surface.

Book on an inclined surface; the slanted surface pushes back normal to the slanted surface.

How many types of Forces have we seen so far?

• Applied Force – “pushing/pulling” by hand.

• Gravitational Force (or “weight”) = mg

• Tension – applied by a rope or string

• Frictional force – applied by a surface to an object, acting parallel to the surface.

• Normal Force applied by a surface to an object, acting normal to the surface.

What Forces act on a Balloon Pressed Against a Wall?

What would a Free Body Diagram of the balloon look like?

People, Scales and Tables

• Suppose I stand on a Scale while pushing upwards on a table. How will the scale read?– If I push up on table, it pushes down on me.– Greater than my weight. Do a Free Body Diagram.

• Suppose I stand on a Scale while pushing down on a table. How will the scale read?– If I push down on table, it pushes up on me.– Less than my weight. Do a Free Body Diagram.

Can the Third Law cause Objects to Move?

• Yes. Examples:– I can push myself off the wall.– A balloon forces air out one end; the air forces balloon

forward.– Car thrusting air one way is propelled the other way.

• How does a car move? Motion is transmitted from:– Engine to Transmission to Wheels– Wheels exert a backward force on the road– The road exerts an equal and opposite forward force

on car, moving it forward.

The Third Law, The Second Law, and a Couple of Carts.

• A cart hits a stationary cart of the same mass. How do you expect them to move after the collision?

• A cart hits a heavier stationary cart. How do you expect them to move after the collision?

• A cart hits a lighter stationary cart. How do you expect them to move after the collision?

The Horse Cart Problem

The Horse-Cart Problem (cont’d)

• Let’s look at the forces– On the Horse ONLY– On the Cart ONLY– On the Horse and Cart taken as ONE OBJECT

Forces Involved• The “Action Forces”:

Let P = Force by Horse Pulling on Cart (Right)Let F = Force by Horse Pushing on Ground (Left)Let f = friction by Ground acting on Cart (Left)

• What are the Reaction Forces to these Action Forces?P = Force by Cart pulling back on Horse (Left)F = Force by Ground pushing back on Horse (Right) f= friction force by Cart on Ground (Right)

f P P F f F

FNET = P – f = MC a

FNET = F – P = MH a

FNET = F – f = (MC+MH) a

FBD on Cart FBD on Horse FBD on Horse + Cart

Forces Involved

• Starting from Rest, to get moving:– The Horse must __________ on the cart to overcome _________.– The Horse must ________ on the ground to overcome ________.

• How are the FBD’s on the Cart and Horse equivalent to the FBD on both?

• Once moving what must the horse do to keep moving at constant velocity?

f P

FNET = P – f = MC a

FBD on Cart

f F

FNET = F – f = (MC+MH) a

P F

FNET = F – P = MH a

FBD on Horse FBD on Horse + Cart

Numerical Problem

• Mass of Cart = 800 kg

• Mass of Horse = 400 kg

• Force of Friction = 1000 N

• With what force must the horse pull on the cart in order to move the cart?

• If the Horse pushes on the ground with a force of 1600 N what will be the acceleration?

• With what force must the horse push on the ground to stay at constant speed?

Cannon and Cannonball (or “Ball”)

A 500 kg cannon fires a 20 kg ball East, exerting a force on the ball of 1000N for 2 seconds.

1)What force is exerted by the ball back on the cannon?

2)What is the acceleration of the ball?

3)What is the acceleration of the cannon?

4)What is the final velocity of the ball (vB)?

5)What is the final velocity of the cannon(vC)?

6)Signs and Portents:

a) What is mBvB?

b) What is mCvC?

A Closer Look at Friction

• What causes friction between two surfaces?– Surface Characteristics (Roughness, Stickiness)

– Force with which surfaces are pressed together.

• We specify these properties by:– μ = measure of surface roughness or stickiness

– N = Normal Force between the surfaces.

• Static Friction (fs): equals sum of other forces exerted parallel to surface until a threshold is reached where:

fs = μsN. So we say: fs ≤ μs N

• Kinetic Friction (fk): Once you get moving it’s easier to stay moving: fk = μk N ; μk < μs

• μk , μs : “Coefficients of static or kinetic friction.”