Lecture 7 Newton ’ s Laws of Motion

Lecture 7

Newton’s Laws of Motion

Midterm Test #1 - Thursday!

- 21 multiple-choice problems- A calculator will be needed.- CHECK YOUR BATTERIES!- NO equations or information may be stored in your calculator. This is part of your pledge on the exam.- Scratch paper will be provided, to be turned in at the end of the exam.- A sign-in sheets will be used, photographs of the class will be taken, all test papers will be collected

Ftop

FbotW

A weight on a string...if I pull the bottom string down, which string will break first?

a) top string

b) bottom string

c) there is not enough information to answer this question

How quickly is the string pulled? A sudden, strong tug is resisted by the inertia of the mass, protecting the top string. A gradual pull forces the top string to keep the system in equilibrium.

Translation Equilibrium

“translational equilibrium” = fancy term for not accelerating = the net force on an object is zero

example: book on a table

example: book on a table in an elevator at constant velocity

Before practicing his routine on the rings, a 67-kg gymnast stands motionless, with one hand grasping each ring and his feet touching the ground. Both arms slope upward at an angle of 24° above the horizontal.

(a) If the force exerted by the rings on each arm has a magnitude of 290 N, and is directed along the length of the arm, what is the magnitude of the force exerted by the floor on his feet?

(b) If the angle his arms make with the horizontal is greater that 24°, and everything else remains the same, is the force exerted by the floor on his feet greater than, less than, or the same as the value found in part (a)? Explain.

Before practicing his routine on the rings, a 67-kg gymnast stands motionless, with one hand grasping each ring and his feet touching the ground. Both arms slope upward at an angle of 24° above the horizontal.

(a) If the force exerted by the rings on each arm has a magnitude of 290 N, and is directed along the length of the arm, what is the magnitude of the force exerted by the floor on his feet?

(b) If the angle his arms make with the horizontal is greater that 24°, and everything else remains the same, is the force exerted by the floor on his feet greater than, less than, or the same as the value found in part (a)? Explain.

N

a)

b) if the angle is larger and everything else remains the same, the applied forces are more vertical. With more upward force from the arms, LESS normal force is required for zero acceleration

Lecture 6

Applications of Newton’s Laws

(Chapter 6)

8

a) N > mg

b) N = mg

c) N < mg (but not zero)

d) N = 0

e) depends on the size of the elevator

m a

A block of mass m rests on

the floor of an elevator that is

accelerating upward. What is

the relationship between the

force due to gravity and the

normal force on the block?

Going Up II

9

The block is accelerating upward, so it

must have a net upward force. The

forces on it are N (up) and mg (down), so

N must be greater than mg in order to

give the net upward force!

a) N > mg

b) N = mg

c) N < mg (but not zero)

d) N = 0

e) depends on the size of the elevator

F = N – mg = ma > 0Thus, N =mg+ma > mg

m a > 0

mg

N

A block of mass m rests on

the floor of an elevator that is

accelerating upward. What is

the relationship between the

force due to gravity and the

normal force on the block?

Going Up II

Follow-up: What is the normal force if the elevator is in free fall downward?

10

You are holding your 2.0 kg

physics text book while

standing on an elevator.

Strangely, the book feels as if

it weighs exactly 2.5 kg. From

this, you conclude that the

elevator is:

Elevate Mea) in freefall

b) moving upwards with a constant velocity of 4.9 m/s

c) moving down with a constant velocity of 4.9 m/s

d) experiencing a constant acceleration of about 2.5 m/s2 upward

e) experiencing a constant acceleration of about 2.5 m/s2 downward

10

Use Newton’s 2nd law! the apparent weight:

You are holding your 2.0 kg

physics text book while

standing on an elevator.

Strangely, the book feels as if

it weighs exactly 2.5 kg. From

this, you conclude that the

elevator is:

Elevate Mea) in freefall

b) moving upwards with a constant velocity of 4.9 m/s

c) moving down with a constant velocity of 4.9 m/s

d) experiencing a constant acceleration of about 2.5 m/s2 upward

e) experiencing a constant acceleration of about 2.5 m/s2 downward

and the sum of forces:

give a positive acceleration ay

12

Frictional Forces

Friction has its basis in surfaces that are not completely smooth:

Kinetic friction

Kinetic friction: the friction experienced by surfaces sliding against one another

The frictional force is proportional to the contact force between the two surfaces (normal force):

The constant is called the coefficient of kinetic friction.

fk always points in the direction opposing motion of two surfaces

Frictional ForcesFrictional Forces

fk

fk

Naturally, for any frictional force on a body, there is an opposing reaction force on the other body

Frictional ForcesFrictional Forces

fk

fk

when moving, one bumps “skip” over each other

fs

fs

when relative motion stops, surfaces settle

into one another

static friction

The static frictional force tries to keep an object from starting to move when other forces are applied.

Static Friction

The static frictional force has a maximum value, but may take on any value from zero to the

maximum... depending on what is needed to keep the sum of forces to zero.

The maximum static frictional force is also proportional to

the contact force

A block sits on a flat table. What is the force of static friction?

Static Friction

a) zero

b) infinite

c) you need to tell me stuff, like the mass of the block, μs, and what planet this is happening on

Characteristics of Frictional Forces• Frictional forces always oppose relative motion

•Static and kinetic frictional forces are independent of the area of contact between objects

• Kinetic frictional force is also independent of the relative speed of the surfaces.

(twice the mass = twice the weight = twice the normal force = twice the frictional force)

• Coefficients of friction are independent of the mass of objects, but in (most) cases forces are not:

Coefficients of Friction

Q: what units?

20

Measuring static coefficient of friction

N

W

fs

x

y

Wx

Wy

If the block doesn’t move, a=0.

at the critical point

Given the “critical angle” at which the block starts to slip, what is μs?

21

Acceleration of a block on an incline

N

W

fk

x

y

Wx

Wy

If the object is sliding down -

v

22

Acceleration of a block on an incline

N

W

fk

x

y

Wx

Wy

If the object is sliding up -

v

What will happen when it stops?

23

m

a) not move at all

b) slide a bit, slow down, then stop

c) accelerate down the incline

d) slide down at constant speed

e) slide up at constant speed

A mass m is placed on an inclined plane ( > 0) and slides down the plane with constant speed. If a similar block (same ) of mass 2m were placed on the same incline, it would:

Sliding Down II

24

The component of gravity acting down

the plane is double for 2m. However,

the normal force (and hence the

friction force) is also double (the same

factor!). This means the two forces

still cancel to give a net force of zero.

A mass m is placed on an inclined plane ( > 0) and slides down the plane with constant speed. If a similar block (same ) of mass 2m were placed on the same incline, it would:

W

Nf

Wx

Wy

a) not move at all

b) slide a bit, slow down, then stop

c) accelerate down the incline

d) slide down at constant speed

e) slide up at constant speed

Sliding Down II

25

Tension

When you pull on a string or rope, it becomes taut. We say that there is tension in the string.

Note: strings are “floppy”, so force from a string is along the string!

26

Tension in a chainTension in a chain

W

Tup

Tdown

Tup = Tdown when W = 0

In this class: we will assume that all ropes, strings, wires, etc. are massless unless otherwise stated.

Tension is the same everywhere in a massless rope!

27

Massive vs. Massless Rope

The tension in a real rope will vary along its length, due to the weight of the rope.

In this class: we will assume that all ropes, strings, wires, etc. are massless unless otherwise stated.

T1 = mg

m

T3 = mg + Wr

T2 = mg + Wr/2

Tension is the same everywhere in a massless rope!

28

Three Blocks

T3 T2 T13m 2m m

a

a) T1 > T2 > T3

b) T1 < T2 < T3

c) T1 = T2 = T3

d) all tensions are zero

e) tensions are random

Three blocks of mass 3m, 2m, and

m are connected by strings and

pulled with constant acceleration a.

What is the relationship between

the tension in each of the strings?

29

T1 pulls the whole set

of blocks along, so it

must be the largest.

T2 pulls the last two

masses, but T3 only

pulls the last mass.

Three Blocks

T3 T2 T13m 2m m

a

a) T1 > T2 > T3

b) T1 < T2 < T3

c) T1 = T2 = T3

d) all tensions are zero

e) tensions are random

Three blocks of mass 3m, 2m, and

m are connected by strings and

pulled with constant acceleration a.

What is the relationship between

the tension in each of the strings?

Follow-up: What is T1 in terms of m and a?

30

TensionTension

Force is always along a rope

W

TTTy

TT

31

Idealization: The PulleyAn ideal pulley is one that simply changes the

direction of the tension

distance box moves = distance hands move

speed of box = speed of hands

acceleration of box = acceleration of hands

32

Tension in the rope?

33

2.00 kg

Tension in the rope?

W

W

TT

34

fk

y :m1 : x :

m2 : y :

μk

μkμk

μk

μk

35

Over the Edge

m

10 kg a

m

a

F = 98 N

Case (1) Case (2)

a) case (1)

b) acceleration is zero

c) both cases are the same

d) depends on value of m

e) case (2)

In which case does block m experience

a larger acceleration? In case (1) there

is a 10 kg mass hanging from a rope

and falling. In case (2) a hand is

providing a constant downward force of

98 N. Assume massless rope and

frictionless table.

35

36

In case (2) the tension is

98 N due to the hand.

In case (1) the tension is

less than 98 N because

the block is accelerating

down. Only if the block

were at rest would the

tension be equal to 98

N.

Over the Edge

m

10 kg a

m

a

F = 98 N

Case (1) Case (2)

a) case (1)

b) acceleration is zero

c) both cases are the same

d) depends on value of m

e) case (2)

In which case does block m experience

a larger acceleration? In case (1) there

is a 10 kg mass hanging from a rope

and falling. In case (2) a hand is

providing a constant downward force of

98 N. Assume massless rope and

frictionless table.

36

37

Springs

Hooke’s law for springs states that the force increases with the amount the spring is stretched or compressed:

The constant k is called the spring constant.

38

SpringsNote: we are discussing the force of the spring on the mass. The force of the spring on the wall are equal, and opposite.

39

Springs and TensionA mass M hangs on spring 1, stretching it length L1

Mass M hangs on spring 2, stretching it length L2

Now spring1 and spring2 are connected end-to-end, and M1 is hung below. How far does the combined spring stretch?

S1

S2

a) (L1 + L2) / 2b) L1 or L2, whichever is smallerc) L1 or L2, whichever is biggerd) depends on which order the springs are attachede) L1 + L2

40




S1

S2

W

Fs=T


41




Spring 1 supports the weight.Spring 2 supports the weight.Both feel the same force, and stretch the same distance as before.

S1

S2

W

Fs=T


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Lecture 7 Newton ’ s Laws of Motion