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Lecture 3

NEWTONS LAWS

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Lecture 3

NEWTONS LAWS

1 Newtons Laws

2 Solving Problems

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Lecture 3

NEWTONS LAWS

1 Newtons Laws

2 Solving Problems

3 Examples

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Newtons Laws: First Law

I

Newtons Laws Newtons Laws 2/9

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Newtons Laws: First Law

I Every body perseveres in its state of rest or of

uniform motion in a right line unless it iscompelled to change that state by forcesimpressed thereon.

Newtons Laws Newtons Laws 2/9

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Newtons Laws: First Law

I Every body perseveres in its state of rest or of

uniform motion in a right line unless it iscompelled to change that state by forcesimpressed thereon.

Law of InertiaDefinition of an Inertial Frame:

Frame in which a free body (not acted upon by

net external force) has a constant velocity

(could be zero)

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Newtons Laws: First Law

I Every body perseveres in its state of rest or of

uniform motion in a right line unless it iscompelled to change that state by forcesimpressed thereon.

Law of InertiaDefinition of an Inertial Frame:

Frame in which a free body (not acted upon by

net external force) has a constant velocity

(could be zero)

Inertial Frames Exist.

Newtons Laws Newtons Laws 2/9

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Newtons Laws: Second Law

II

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res = ma

S

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res = ma

CAUSE

N L S d L

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res = ma

CAUSE EFFECT

N t L S d L

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res = ma

F causes change in motion:

N t L S d L

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Newtons Laws: Second Law

II The alteration of the quantity of motion is everproportional to the motive force impressed and

is made in the direction of the right line inwhich that force is impressed.

The net force on a body is the rate of change of

its momentum:F = dp

dtwhere p mv.

For a point particle,dp

dt= ma.

N

i=1

F i =

F res = ma

F causes change in motion: ma is NOT a force!

Newtons Laws Newtons Laws 3/9

Ne tons La s Third La

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Newtons Laws: Third Law

III

Newtons Laws Newtons Laws 4/9

Newtons Laws: Third Law

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Newtons Laws: Third Law

III To every action there is always an equal

reaction or the mutual actions of two bodiesupon each other are always equal & directed tocontrary parts.

Newtons Laws Newtons Laws 4/9

Newtons Laws: Third Law

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Newton s Laws: Third Law

III To every action there is always an equal

reaction or the mutual actions of two bodiesupon each other are always equal & directed tocontrary parts.

F 12 =

F 21

Newtons Laws Newtons Laws 4/9

Newtons Laws: Third Law

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Newton s Laws: Third Law

III To every action there is always an equal

reaction or the mutual actions of two bodiesupon each other are always equal & directed tocontrary parts.

F 12 =

F 21

Mutual forces generally along the line joining theparticles but not always

Newtons Laws Newtons Laws 4/9

Newtons Laws: Third Law

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Newton s Laws: Third Law

III To every action there is always an equal

reaction or the mutual actions of two bodiesupon each other are always equal & directed tocontrary parts.

F 12 =

F 21

Mutual forces generally along the line joining theparticles but not always

i

j=i

F ij = 0

Newtons Laws Newtons Laws 4/9

Newtons Laws: Third Law

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Newton s Laws: Third Law

III To every action there is always an equal

reaction or the mutual actions of two bodiesupon each other are always equal & directed tocontrary parts.

F 12 = F 21Mutual forces generally along the line joining theparticles but not always

i

j=i

F ij = 0

Mutual (internal) Forces in a system of particlescancel

Newtons Laws Newtons Laws 4/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Decide and fix Reference Frame

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Decide and fix Reference Frame Lab Frame: Inertial

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Decide and fix Reference Frame Lab Frame: Inertial Frame fixed to a moving body (could be accelerating!): Non-inertial

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Decide and fix Reference Frame Lab Frame: Inertial Frame fixed to a moving body (could be accelerating!): Non-inertial

Newtons II law not valid in NI frames!

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Solving Problems involving Newton s Laws

Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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g g

Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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g g

Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass Resolve along frame axis directions

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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g g

Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass Resolve along frame axis directions

Write Newtons IInd law for each mass in each direction

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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g g

Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass Resolve along frame axis directions

Write Newtons IInd law for each mass in each direction

Note Constraint equations if any

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass Resolve along frame axis directions

Write Newtons IInd law for each mass in each direction

Note Constraint equations if any

Should be as many eqns as unknowns

Newtons Laws Solving Problems 5/9

Solving Problems involving Newtons Laws

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Decide and fix Reference Frame Lab Frame: Inertial

Frame fixed to a moving body (could be accelerating!): Non-inertialNewtons II law not valid in NI frames!

Draw Free Body Diagrams (FBD) for each mass Identify forces on each mass Resolve along frame axis directions

Write Newtons IInd law for each mass in each direction

Note Constraint equations if any

Should be as many eqns as unknowns

Algebraic soln of unknowns

Newtons Laws Solving Problems 5/9

Example 1: Sliding Blocks

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Example 1: Sliding Blocks

Fm2

m1

= 0

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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Example 1: Sliding Blocks

What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Fm2

m1

= 0

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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a p e S d g oc s

What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Fm2

m1

= 0

Step 1: Draw Free Body

Diagram

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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p g

What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

m1

fs

N

m1g

Step 1: Draw Free Body

Diagram

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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p g

What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

m2

F

m2g

fs

N N

Step 1: Draw Free Body

Diagram

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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p g

What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

m2

F

m2g

fs

N N

Step 1: Draw Free Body

Diagram

Step 2: Apply II Law

(horizontal direction)

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

m1

fs

N

m1g

Step 1: Draw Free Body

Diagram

Step 2: Apply II Law

(horizontal direction)

For m1 : fs = m1a1

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

m2

F

m2g

fs

N N

Step 1: Draw Free Body

Diagram

Step 2: Apply II Law

(horizontal direction)

For m1 : fs = m1a1

For m2 : F fs = m2a2

Newtons Laws Examples 6/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a = Fm1 + m2

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a = Fm1 + m2

fs = m1a = Fm1

m1 + m2

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a = Fm1 + m2

fs = m1a = Fm1

m1 + m2 N= m1g

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a = Fm1 + m2

fs = m1a = Fm1

m1 + m2 N= m1g

=Fmax = (m1 + m2)g

Newtons Laws Examples 7/9

Example 1: Sliding Blocks

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What is the maximum force that can be applied on m2 so m1 does not

slide wrt m2?

Condition: No sliding:

= a1 = a2 = a = Fm1 + m2

fs = m1a = Fm1

m1 + m2 N= m1g

=Fmax = (m1 + m2)g

Qn: What are a1 and a2 if F

Fmax?

Newtons Laws Examples 7/9

Example 2: Block sliding down an accelerating wedge

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1h L

m

M

A

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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g p g

m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1h L

m

M

A

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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g p g

m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1h L

Solution 1: Inertial Frame

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X h Lx

y

Solution 1: Inertial Frame

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Note:

N = mg cos

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

lid d (f i i ) Fi d i l i

http://find/http://goback/

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Note:

N = mg cos A does not figure in (1)

and (2).

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

lid d (f i ti ) Fi d it l ti

http://find/http://goback/

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Note:

N = mg cos A does not figure in (1)

and (2).

Constraint: xX= (h y)cot

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

lid d (f i ti ) Fi d it l ti

http://find/http://goback/

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Note:

N = mg cos A does not figure in (1)

and (2).

Constraint: xX= (h y)cot = xA = y cot (3) .

Newtons Laws Examples 8/9

Example 2: Block sliding down an accelerating wedge

45 wedge M is pushed along a table with acceleration A. Small block

slides on wedge (friction ) Find its acceleration

http://find/http://goback/

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m slides on wedge (friction ). Find its acceleration.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X hN

mg

N

L

x

y

Solution 1: Inertial Frame

mx = Nsin + Ncos (1)

my = Ncos

Nsin

mg (2)

Note:

N = mg cos A does not figure in (1)

and (2).

Constraint: xX= (h y)cot = xA = y cot (3) .

Solving, y =1

2[(1 )A (1 + )g]

Newtons Laws Examples 8/9

Solution 2: Non-intertial Frame

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N + A i

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N + A i

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

(No motion along vertical in this

frame)

Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N mg cos + mA sin

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

(No motion along vertical in this

frame)

ma = mg sin N+ mA cos (up the incline)

Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N = mg cos + mA sin

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

(No motion along vertical in this

frame)

ma = mg sin N+ mA cos (up the incline)

Use = 45

Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N = mg cos + mA sin

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

(No motion along vertical in this

frame)

ma = mg sin N+ mA cos (up the incline)

Use = 45

Solving, a =1

2[(1 + )g + (1 )A]

Newtons Laws Examples 9/9

N

Solution 2: Non-intertial Frame

N = mg cos + mA sin

http://find/http://goback/

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mg N mAN= mg cos + mA sin

(No motion along vertical in this

frame)

ma = mg sin N+ mA cos (up the incline)

Use = 45

Solving, a =1

2[(1 + )g + (1 )A]

y = a sin , giving the same answer as before.

Newtons Laws Examples 9/9

http://find/http://goback/