Physics of MotionLecturer: Mauro Ferreira

Text book: University Physics (extended version with modern Physics) by Young and Freedman, Addison-Wesley, 9th edition.

Before we start, please remember to switch off

your mobile phone

Objectives:By the end of this lecture you should be able to:

• Identify centripetal forces in circular motion• Include air resistance in the motion of an

object • Solve problems about the topics above

Outline• 2nd Newton’s law for curvilinear

motion + problems

• Let’s include air resistance

X

Y

Uniform circular motion is the motion of a particle that moves in a circle with constant speed.For instance, a car rounding a curve with constant radius, a satellite in a circular orbit or a skater moving in a circle, all with constant speed.

aV

The so-called centripetal acceleration is perpendicular to the velocity and always points towards the centre of the circular trajectory.

X

Y

1r

2r

1V

2V

Let’s select a pair of points in the circle for which their position vectors differ by an angle .

Lecture 5

1r

2rr

Let’s select a pair of points in the circle for which their position vectors differ by an angle .

1V

2V

V

Since r1=r2=R and V1=V2=V, the corresponding velocities form an equivalent triangle.

tRrV

t

VRr

V

V

From basic geometry, we have that

In the limit when t0

RVa2

Lecture 5

X

Y

aV

Therefore, the centripetal acceleration is inversely proportional to the circular radius and directly proportional to the velocity squared.

RVa2

Lecture 5

But according to 2nd law, acceleration is the result of a net force

Centripetal acceleratio

n

Centripetal force

V

cF

cc amFamR

RVmFc2

Consider a car of mass M moving at speed V along a circular path of radius R. Identify the forces on the vehicle.

R

Front view

W

N

F

W

N

F

RVMF

MgWNMgW

2

The magnitude of the forces are:

Assuming that the (static) friction coefficient is , what is the maximum speed the car can have to complete the curve in safety ?

RVMgM

NF2max

RgV max

Vmax is mass independent !!!

W

N

F

RgV max

Can you estimate the maximum safe speed of a car driving on a road ?

Estimates:

mRsmg

100/10

12

smV /32100101max

hkmV /115max

Banked curves are built to reduce the risk of cars skidding off the road. Let’s see why:

W

N

Banked curves are built to reduce the risk of cars skidding off the road. Let’s see why:

W

yN

xN

The normal component Nx acts as the centripetal force

cossin

NNNN

y

x

MgN y RVMN x2

RgV 2

tan Cars can round a curve with no friction

T

A sphere of mass 0.5kg is attached to the end of a 1.5m-long cord. The sphere is whirled in a horizontal circle, as shown in the figure. Assuming that the cord can withstand a maximum tension of 50N, what is the maximum speed at which the sphere can be whirled before the cord breaks ?

RVmT2

mRTV max

max

smV /2.12max

A small sphere of mass m is attached to the end of a cord of length R and set in motion in a vertical circle, as shown in the figure. Determine the tension in the cord when the speed of the particle is V and the cord makes an angle with the vertical.

T

W

A small sphere of mass m is attached to the end of a cord of length R and set in motion in a vertical circle, as shown in the figure. Determine the tension in the cord when the speed of the particle is V and the cord makes an angle with the vertical.

T

yW

xW

RVmWT y

2

sin

cos

mgW

mgW

x

y

cos

2

gRVmT

What if the sphere is in the top part of the circular trajectory ? In this case the tension is given by

T

g

RV

mT top2

W

RgVtop Note that the tension vanishes for . Discuss what happens for V<Vtop .

A child being pushed on a swing reaches increasingly larger heights. If the swing is not able to support the load, where will it break ?

cos

2

gRVmT

T is maximum when =0.

Conical pendulum: A small object of mass m suspended from a string of length L revolves with constant angular speed forming an angle with the vertical direction, as shown in the figure. Find an expression for the angular speed

T

W

xT

W

yT

cossinTTTT

y

x

sin2LmT

mgT

x

y

sintan 2Lmmg

Conical pendulum: A small object of mass m suspended from a string of length L revolves with constant angular speed forming an angle with the vertical direction, as shown in the figure. Find an expression for the angular speed

cosLg

A single bead can slide with negligible friction on a wire that is bent into a circular loop of radius R, as shown in the figure. The circle is in a vertical position and rotates steadily around its vertical diameter with angular speed . Determine the angle .

W

N

W

yN

xN

cossin

NNNN

y

x

RVmN

mgN

x

y

2

sin)sin(tan2

RRmmg

Rg2cos

A single bead can slide with negligible friction on a wire that is bent into a circular loop of radius R, as shown in the figure. The circle is in a vertical position and rotates steadily around its vertical diameter with angular speed . Determine the angle .

Is air resistance always

negligible ?

We have avoided the effect of air resistance so far, but what happens when it cannot be neglected? For objects moving in a fluid, a resistance force arises opposing the motion. This force depends on the velocity and is generally given by:

VkR

Interpret the equation with you own words

For a body of mass m falling vertically, we have

dtdVmkVmg

kmgV

mk

dtdV

tV

dtmk

kmgV

dV

00T

V Air resistanceFree fall

mktekmgtV 1)(

It is instructive to see what happens when k0.Terminal velocity is VT=mg/k.

W

0t

aR

1t 2t 3t 4t The physical interpretation of the VxT graph is very simple. As the object starts moving, the velocity increases and so does the air resistance up to the point where Ra is as large as the weight. At this point the velocity no longer changes reaching its terminal value.

We see that the air resistance is not proportional to the mass of the object. That’s why heavier objects tend to fall faster.

mkt

mktt

mkt

egkmt

kmgStS

ekmgdttS

ekmgtV

2

0

0

)(

1)(

1)(

What about the displacement ?

Once again, it is instructive to see what happens when k0. 2

2tgS

Objectives:By the end of this lecture you should be able to:

• Identify centripetal forces in circular motion• Include air resistance in the motion of an

object • Solve problems about the topics above

Puzzles & challengesE-mail: ferreirm@tcd.ie

R = ?

A projectile launched with speed V0 at an angle with the horizontal lands on an incline of angle (see figure). Find the distance R in terms of the relevant quantities. What is the optimum angle opt for which the distance R is maximum ?

Assignments & tutorials

Tutorial sheet to be handed in on: 17/11/2003

before 12:00pm