PHYSICS 231

INTRODUCTORY PHYSICS I

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 neighbors or the medium

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

Sliding Friction

• Parallel to surface, opposing direction of motion

• ~ independent of the 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 to keep object at rest.

• s is coefficient of static friction

• N is the normal force

f

F

fs ≤sN

Kinetic Friction, ƒk

• k is coefficient of kinetic friction

• Friction force opposes direction of motion

• N is the normal force F

f€

fk = μ kN

€

k < μ s

Coefficients of Friction

f ≤sNf =kN

s > k

Example 4.7

The man pushes/pulls with a force of 200 N. Thechild and sled combo has a mass of 30 kg and the coefficient 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 s would stop the block from sliding?b) If the box is sliding and k = 0.2, what is the acceleration?c) What is the tension of the rope?a) s = 0.5 b) a=1.96 m/s2 c) 39.25 N

Example 4.9

What is the minimum s required to prevent a sled from slipping down a hill of slope 30 degrees?

s = 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 ⋅(

rxf −

rxi )

=FΔxcosθ

Units of Work and Energy

SI unit = Joule1 J = 1 Nm = 1 kgm2/s2

W =F ⋅x

Example 5.0

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

120 m

Work can be positive or negative

• Man does positive work lifting box

• Man does negative work lowering box

• Gravity does positive work when box lowers

• Gravity does negative work when box is raised

Work and friction

A block is pulled a distance Δx by 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=aΔx

KE =12

mv2

Work-Energy Theorem

€

KE f −KE i = FnetΔx

€

KE f −KE i =Wnet€

12mv f

2 − 12mv i

2 = maΔx

Example 5.1

A skater of mass 60 kg has an initial velocity of 12 m/s. He slides on ice where the frictional force is 36 N. 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)

ΔPE = −FΔx

ΔPE = 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 +KE f =PEi +KEi

ΔKE =−ΔPE

Example 5.2

A diver of mass m drops from a board 10.0 m above the water surface, as in the Figure. Find his speed 5.00 m above 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 are thrown from the top of a building with the same initial speed. Initially, Ball 1 moves horizontally. Ball 2 moves upward. Ball 3 moves downward.

Neglecting air resistance, which ball has the fastest speed 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), are connected 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 and incline are frictionless.

s

Example 5.5

1.51 m/s

Example 5.6

Tarzan swings from a vine whose length is 12 m. If Tarzan starts at an angle of 30 degrees with respect to the vertical and has no initial speed, what is his speed at the bottom of the arc?

5.61 m/s