Lecture 7

Applications of Newton’s Laws

(Chapter 6)

Reading and Review

Going Up I

A block of mass m rests on the floor of

an elevator that is moving upward at

constant speed. What is the

relationship between the force due to

gravity and the normal force on the

block?

a) N > mg

b) N = mg

c) N < mg (but not zero)

d) N = 0

e) depends on the size of the elevator

m

v

The block is moving at constant speed, so

it must have no net force on it. The

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

N = mg, just like the block at rest on a

table.

Going Up I

A block of mass m rests on the floor of

an elevator that is moving upward at

constant speed. What is the

relationship between the force due to

gravity and the normal force on the

block?

a) N > mg

b) N = mg

c) N < mg (but not zero)

d) N = 0

e) depends on the size of the elevator

m

v

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

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 > 0→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?

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.

This 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 Forces

fk

fk

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

Frictional Forces

fk

fk

when moving, one bumps “skip” over each

other

fs

fs

when relative motion stops, surfaces settle

into one another

static frictionkinetic friction

Kinetic Friction ≤ 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

Similarities between→normal forces and→static friction

• Variable; as strong as necessary to prevent relative motion either

- perpendicular to surface (normal)

- parallel to surface (friction)

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?

Going Sledding

1

2

a) pushing her from behind

b) pulling her from the front

c) both are equivalent

d) it is impossible to move the sled

e) tell her to get out and walk

Your little sister wants

you to give her a ride

on her sled. On level

ground, what is the

easiest way to

accomplish this?

Going Sledding

1

2

In case 1, the force F is pushing down

(in addition to mg), so the normal

force is larger. In case 2, the force F

is pulling up, against gravity, so the

normal force is lessened. Recall that

the frictional force is proportional to

the normal force.

a) pushing her from behind

b) pulling her from the front

c) both are equivalent

d) it is impossible to move the sled

e) tell her to get out and walk

Your little sister wants

you to give her a ride

on her sled. On level

ground, what is the

easiest way to

accomplish this?

Measuring static coefficient of friction

N

W

fs

x

y

Wx

Wy

If the block doesn’t move, a=0.

at the critical point

Acceleration of a block on an incline

N

W

fk

x

y

Wx

Wy

If the object is sliding down -

v

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?

A mass m, initially moving with a speed of 5.0 m/s, slides up a 30o ramp. If the coefficient of kinetic friction is 0.4, how far up the ramp will the mass slide?If the coefficient of static friction is 0.6, will the mass eventually slide down the ramp?

A mass m, initially moving with a speed of 5.0 m/s, slides up a 30o ramp. If the coefficient of kinetic friction is 0.4, how far up the ramp will the mass slide?If the coefficient of static friction is 0.6, will the mass eventually slide down the ramp?

2

2 20 0

2 20 0

?

ˆ : 0 cos cos

ˆ : sin sin cos

sin cos sin cos

9.81 sin30 0.4cos30 8.30

0 2

2 5.0 2 8.30 1.51

When mass stops:

s

x k k

xx k k

o o

f x f

f x

x

y N mg N mg

x F mg N mg mg

Fa g g gm

m s

v v a x x

x x x v a m

F mg

in sin cos

sin cos sin30 cos30

0.02 0

mass will remain at rest

s s

o os s

N mg mg

mg mg

mg

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

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

Translational 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

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!

Tension in a chain

W

Tup

Tdown

Tup = Tdown when W = 0

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!

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

direction of the tension

along with a rope: useful class of problems of combined

motiondistance box moves = distance hands move

speed of box = speed of hands

acceleration of box = acceleration of hands

Tension in the rope?

Translational equilibrium?

2.00 kg

Tension in the rope?

Translational equilibrium?

W

W

TT

y :m1 : x :

m2 : y :

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?

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?

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 ropes.

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 ropes.

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