DYNAMICS

1. Newton’s Three Laws

Newton’s First Law Existence of inertial systems of reference

In inertial system of reference, any object acted by no net force remains at rest or continues its motion along straight line with constant velocity

Newton’s Second Law

BAAB FF

Newton’s Third Law

amF

2/11 smkgNF

Note: these two forces act on different objects. Never add these forces!

Units: (1 lb = 4.448 N)1

1.1 Newton’s First Law (examples)

Example 1 (Snapped string): A small ball attached to the end of a string moves in circles as shown below. If the string snaps, what will be the trajectory of the ball?

C

A B

In inertial system of reference, any object acted by no net force remains at rest or continues its motion along straight line with constant velocity

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Example 2: 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 inertiaB) there are no forces acting on it at allC) it does move, but too slowly to be seenD) there is no net force on the bookE) there is a net force, but the book is too heavy to move

There are forces acting on the book, but the only forces acting are in the y-direction. Gravity acts downward, but the table exerts an upward force that is equally strong, so the two forces cancel, leaving no net force.

Example 3: A hockey puck slides on ice at constant velocity. What is the net force acting on the puck?A) more than its weightB) equal to its weightC) less than its weight but more than zeroD) depends on the speed of the puckE) zero

The puck is moving at a constant velocity, and therefore it is not accelerating. Thus, there must be no net force acting on the puck.

Follow-up: Are there any forces acting on the puck? What are they?3

Example 1: Car brakes provide a force F for 5 s. During this time, the car moves 25 m, but does not stop. If the same force would be applied for 10 s, how far would the car have traveled during this time?

1) 100 m2) 50 m < x < 100 m3) 50 m4) 25 m < x < 50 m5) 25 m

In the first 5 s, the car has still moved 25 m. However, since the car is slowing down, in the next 5 s, it must cover less distance. Therefore, the total distance must be more than 25 m but less than 50 m.

?

10

5

25

2

2

1

1

d

st

st

md

1

211

21

10

22

202

21

101

12

222

22

2

2

2

datdta

tvat

tvd

attvd

tt

mFa Acceleration:

1.2 Newton’s Second Law (examples)

amF

4

F

a1

m1 F

a2 = 2a1

m2

If both masses are put together and the same force is applied to the combination, what is the resulting acceleration?

a ?

m2F m1

A. 2/3 a1

B. 3/2 a1

C. 3/4 a1

Example 2: A force F acting on a mass m1 results in an acceleration a1.

The same force acting on a mass m2 results in an acceleration a2 = 2a1.

11 2

1 2 1

111 2 1 1 12

mF F Fm = m = = =

a a 2a 2

F F 2 F 2a = = = a

m + m m + m 3 m 3

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If object A exerts a force on object B (an “action”), then

object B exerts a force on body A (a “reaction”).

These two forces have the same magnitude but opposite direction. Note: these two forces act on different objects.

1.3 Newton’s Third Law (examples)

For every force, or action force, there is an equal but opposite force, or reaction force.

BAAB FF

A

B

BAF

ABF

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Example: Two carts are put back-to-back on a track. Cart A has a spring-loaded piston; cart B, which has twice the mass of cart A, is entirely passive. When the piston is released, it pushes against cart B, and the carts move apart. Which of the two forces exerted by the two carts on each other has a larger magnitude?

1. The force exerted by A.2. The two forces have equal magnitude.3. The force exerted by B.

AB

It’s a third law pair!!

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Example 1: You tie a rope to a tree and you pull on the rope with a force of 100 N. What is the tension in the rope?

The tension in the rope is the force that the rope “feels” across any section of it (or that you would feel if you replaced a piece of the rope). Since you are pulling with a force of 100 N, that is the tension in the rope.

Application of Newton’s Laws (Ropes and tension)

Example 2: Two tug-of-war opponents each pull with a force of 100 N on opposite ends of a rope. What is the tension in the rope?

This is literally the identical situation to the previous question. The tension is not 200 N !! Whether the other end of the rope is pulled by a person, or pulled by a tree, the tension in the rope is still 100 N !!

Example 3: You and a friend can each pull with a force of 200 N. If you want to rip a rope in half, what is the best way?1) You and your friend each pull on opposite ends of the rope2) Tie the rope to a tree, and you both pull from the same end3) It doesn’t matter -- both of the above are equivalent

Take advantage of the fact that the tree can pull with almost any force (until it falls down, that is!). You and your friend should team up on one end, and let the tree make the effort on the other end.

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Normal force between book and tableNBT = –NTB

WBE

NBT

NTE

NTBWTE

WET WEB

NET

The book does not accelerate WBE+NBT=0

The table does not accelerate WTE+NTB+NTE=0

Does the earth accelerate?

Gravitational force between book and earthWBE = –WEB

Gravitational force between table and earthWTE = –WET

Normal force between table and earth NTE = –NET

Action-Reaction Pairs:

2. Free Body Diagram

Example: Book on Table – The full story

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3. Weight (Force due to gravity)

mgFg

Example: You see two cases: a student pulling or pushing a sled with a force F which is applied at an angle . In which case is the normal force greater?

Case 1Case 1

Case 2Case 2

1) case 12) case 23) it’s the same for both4) depends on the magnitude of the force F5) depends on the ice surface

In Case 1, the force F is pushing down (in addition to mg), so the normal force needs to be larger. In Case 2, the force F is pulling up, against gravity, so the normal force is lessened.

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3a) Apparent weight(The force the body exerts on a support)

Question 1: What can you say about the force of gravity Fg acting on a stone and a feather?

1) Fg is greater on the feather2) Fg is greater on the stone3) Fg is zero on both due to vacuum4) Fg is equal on both always5) Fg is zero on both always

The force of gravity (weight) depends on the mass of the object!! The stone has more mass, therefore more weight.

Question 2: The force of gravity is acting on the stone and the feather falling in vacuum. Which one has greater acceleration?

1) The feather2) The stone3)Both acceleration are zero due to vacuum4)Both acceleration are zero always5)They have the same acceleration

The acceleration is given by F/m The force of gravity (weight) is F=mg, then we end up with acceleration g for both objects.

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Example 1: A box with a mass of 100 kg is given an upward acceleration

of 2.2 m/s² by a cable. What is the tension in the cable?

m

mg

T Newton’s equation: T-mg=ma

T=m(g+a)

T=(100kg)(9.8 m/s2 +2.2 m/s2 )= 1200 N

m=100kga=2.2m/s2

T=?

Note: this tension is bigger than the box’s weight: w=mg=(100 kg)( 9.8 m/s2)= 980 N

Example 2: The same box as in example 1 is given an downward acceleration of 2.2 m/s² by a cable. What is the tension in the cable?

m = 100 kga = -2.2 m/s2

T=?

T=m(g+a)T=(100 kg)(9.8 m/s2 -2.2 m/s2 )= 760 N

Note: this tension is smaller than the box’s weight.Compare examples 1 & 2: the tension depends on acceleration & is independent from velocity. The tension is equal to the weight if there is no acceleration (a=0).

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