PHYSICS 231

INTRODUCTORY PHYSICS I

PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 5

• Range (yi=yf):

• Relative velocity:

• Newton’s Laws:

1. If F = 0, velocity doesn’t change.

2.

3.

Main points of last lecture

R =vi2

gsin2θ

vbe = vbr + vre

rF∑ =m

ra

rF12 =−

rF21

Newton’s Third Law

• Single isolated force cannot exist• For every action there is an equal and opposite reaction

• Action and Reaction Forces act on different objects

Force on “1” due to “2”

rF12 =−

rF21

Free-body Diagram

• Draw arrows for all forces acting ON object

• If the object is in “equilibrium” (at rest or constant v)

• Otherwise, find acceleration by

Ignore rotational motion for now. Treat object as a particle.

rF∑ =0

rF∑ =m

ra

Example 4.1

Two blocks sit on a frictionless table. The masses areM1=2 kg and M2=3 Kg. A horizontal force F=5 N is applied to Block 1.

1. What is the acceleration of the blocks?

2. What is the force of block 1 on block 2?

1. a = 1 m/s2

2. F21= 3 N

M1M2

F

Mechanical Forces

• Gravity: w=mg (down)• Normal forces• Strings, ropes and Pulleys• Friction• Springs (later)

Rules for Ropes and Pulleys

• Force from rope points AWAY from object• (Rope can only pull)

• Magnitude of the force is Tension• Tension is same everywhere in the rope

• Tension does not change when going over pulley

Approximations: Neglect mass of rope and pulley, neglect friction in pulley

Example 4.2

I pull a 5 kg mass up with a rope, so that it accelerates 2 m/s2. What is the tension in the rope?

T = 59 N

Example 4.3 - Atwood Machine

a) Find accelerationb) Find T, the tension in the stringc) Find force ceiling must exert on pulley

a) a = g/6 = 1.635 m/s2

b) T = 57.2 Nc) Fpulley=2T = 114.5 N

5 kg7 kg

Example 4.4a

Which statements are correct?Assume the objects are in static equilibrium.

T1 is _____ T2

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.4b

Which statements are correct?Assume the objects are static.

T2 is ______ T3

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.4c

Which statements are correct?Assume the objects are static.

T3 is ______ Mg

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.4d

Which statements are correct?Assume the objects are static.

T1+T2 is ______ Mg

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.4

Given that Mlight = 25 kg, find all three tensions

T3 = 245.3 N, T1 = 147.4 N, T2 = 195.7 N

Cable Pull Demo

Inclined Planes

• Choose x along the incline and y perpendicular to incline

• Replace force of gravity with its componentsFg,x =mgsinθ

Fg,y =mgcosθ 30.0

Example 4.5

Find the acceleration and the tension

a = 4.43 m/s2, T= 53.7 N

Example 4.6

Find M such that the box slides at constant v

M=15.6 kg

M

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

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

Other kinds of friction

• Air resistance, F ~ Area v2

• Rolling resistance, F ~ v

Terminal velocity:

Fresistance =CAv2

=mgatterminalvelocity

Coffee Filter Demo

Example 4.9

An elevator falls with acceleration a = 8.0 m/s2. If a 200-lb person stood on a bathroom scale during the fall, what would the scale read?

36.9 lbs

Accelerating Reference Frames

• Equivalent to “Fictitious” gravitational force

g fictitious =−aframe

Fictitious Force: Derivation

Eq. of motion in fixed frame

x =v0t+12

at2

=v0t+12

Fm

t2

F-maf looks like force in new frame, maf acts like fake gravitational force!

x0 (t)=12

aft2

x−x0 (t) =v0t+12

(F −maf )m

t2

Example 4.10

You are calibrating an accelerometer so that you can measure the steady horizontal acceleration of a car by measuring the angle a ball swings backwards.If M = 2.5 kg and the acceleration, a = 3.0 m/s2:a) At what angle does the ball swing backwards?b) What is the tension in the string?

θ=17 degT= 25.6 N

θ

Example 4.11a

A fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.In the instant just after the elevator begins to move upward, the reading on the scale will be ______________ 20 lbs. a) Greater than

b) Less thanc) Equal to

Example 4.11b

A fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.

On the way back down, while descending at constant velocity, the reading on the scale will be ________________ 20 lbs.

a) Greater thanb) Less thanc) Equal to

Example 4.11c

A fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.

In the instant just before the elevator comes to a stop on the 78th floor, the mass of the fish will be ______________ 9.072 kg.

a) Greater thanb) Less thanc) Equal to