PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 6

• Gravity• Normal forces• Strings, ropes and Pulleys

• Friction

• Work and Kinetic Energy• Potential Energy• Conservation of Energy

Last Lecture:

Today:

Frictional Forces

• RESISTIVE force between object and neighborsor the medium

• Examples:• Sliding a box• Air resistance• Rolling resistance

Sliding Friction

• Parallel to surface,opposing direction of motion

• ~ independent ofthe area of contact

• Depends on the surfaces in contact• Object at rest: Static friction• Object in motion: Kinetic friction

Static Friction, ƒs

• Just enough force tokeep object at rest.

• µs is coefficient ofstatic friction

• N is the normal force

f

F

fs ! µsN

KineticFriction, ƒk

• µk is coefficient ofkinetic friction

• Friction force opposesdirection of motion

• N is the normal forceF

f!

fk = µkN

!

µk

< µs

Coefficientsof Friction

f ! µsN

f = µkN

µs > µk

Example 4.7

The man pushes/pulls with a force of 200 N. Thechild and sled combo has a mass of 30 kg and thecoefficient of kinetic friction is 0.15. For each case:What is the frictional force opposing his efforts?What is the acceleration of the child?

f=59 N, a=3.80 m/s2 / f=29.1 N, a=4.8 m/s2

Example 4.8

Given m1 = 10 kg and m2 = 5 kg:a) What value of µs would stop the block from sliding?b) If the box is sliding and µk = 0.2, what is theacceleration?c) What is the tension of the rope?

a) µs = 0.5 b) a=1.96 m/s2 c) 39.25 N

Example 4.9

What is the minimum µs required toprevent a sled from slipping down ahill of slope 30 degrees?

µs = 0.577

Chapter 5Work and Energy

Forms of Energy

• Mechanical• Kinetic, gravitational

• Thermal• Microscopic mechanical

• Electromagnetic• Nuclear

Energy is conserved!

Work

• Relates force to change in energy

• Scalar quantity• Independent of time

W =rF ! (

rx f "

rxi )

= F#x cos$

Units of Work and Energy

SI unit = Joule1 J = 1 N⋅m = 1 kg⋅m2/s2

W = F ! x

Example 5.0

A man holds a 50 lb box at waist level for 10minutes. Has he done any work during this time?

120 m

Work can be positive or negative

• Man does positive worklifting box

• Man does negative worklowering box

• Gravity does positivework when box lowers

• Gravity does negativework when box is raised

Work and friction

A block is pulled a distance Δxby constant force F.

• Work of F > 0• Work of N = 0• Work of mg = 0• Work of fk < 0

Work of (kinetic) frictional force is always < 0.-> It removes mechanical energy from system.

N

mg

fk F

Kinetic Energy

Same units as work

Remember the Eq. of motion

Multiply both sides by m,

v f2

2!vi2

2= a"x

KE =1

2mv

2

Work-Energy Theorem

!

KE f "KEi = Fnet#x

!

KE f "KEi =Wnet

!

1

2mv f

2" 1

2mvi

2= ma#x

Example 5.1

A skater of mass 60 kg has an initial velocity of 12m/s. He slides on ice where the frictional force is 36N. How far will the skater slide before he stops?

120 m

Potential Energy

If force depends on distance, we can define Potential Energy

For gravity (near Earth’s surface)

!PE = "F!x

!PE = mgh

This must be independent of Path-> “Conservative Force”

Conservation of Energy

Conservative forces:• Gravity, electrical, …

Non-conservative forces:• Friction, air resistance…

Non-conservative forces still conserve energy!Energy just transfers to thermal energy (heat)

PE f + KEf = PEi + KEi

!KE = "!PE

Example 5.2

A diver of mass m drops froma board 10.0 m above thewater surface, as in theFigure. Find his speed 5.00 mabove the water surface.Neglect air resistance.

9.9 m/s

Example 5.3A skier slides down the frictionless slope as shown.What is the skier’s speed at the bottom?

H=40 m

L=250 m

start

finish

28.0 m/s

Example 5.4

Three identical balls arethrown from the top of abuilding with the same initialspeed. Initially, Ball 1 moves horizontally. Ball 2 moves upward. Ball 3 moves downward.

Neglecting air resistance,which ball has the fastestspeed when it hits the ground?A) Ball 1B) Ball 2C) Ball 3D) All have the same speed.

Two blocks, A and B (mA=50 kg and mB=100 kg), areconnected by a string as shown. If the blocks beginat rest, what will their speeds be after A has slida distance s = 0.25 m? Assume the pulley andincline are frictionless.

s

Example 5.5

1.51 m/s

Example 5.6

Tarzan swings from a vine whoselength is 12 m. If Tarzan starts at anangle of 30 degrees with respect tothe vertical and has no initial speed,what is his speed at the bottom ofthe arc?

5.61 m/s