Angular Motion

Conservation of momentum

• Momentum p:

0F

const.p

Fdt

pd

Example: An object of mass M flies with velocity v splits into two pieces of equal mass M/2 because of the occurrence of an internal explosion. If one piece has velocity u, what is the velocity of the other piece right after the explosion?

vmp

Angular velocity

• Radian(弧度 ): Rl /R

l

3.57180

1

R

v

dt

dl

Rdt

d

1

• Angular speed:

Angular momentum

prL

I

mrmrvL

2

where I is the moment of inertia.

Moment of inertia

3/2mlI

12/2mlI

2/2mrI

A thin rod of length l Through the center

A thin rod of length l Through one end

Sphere of radius r Along a diameter

Cylinder of radius r Along axis of symmetry

5/2 2mrI

Conservation of angular momentum

Fr

dt

pdr

dt

Ld

0 const.L

22

mrωr

mvFc

Centripetal force（向心力）

Centrifugal force?離心力 ?

The simplest angular motion is one in which the body moves along a curved path at a constant velocity, as when a runner travels along a circular path or an automobile rounds a curve.

The usual problem here is to calculate the centripetal forces and determine their effect on the motion of the object.

Force on a Curved Path

A common problem to raise is the maximum speed at which an automobile can round a curve without skidding.

Consider a car of weight W moving on a curved level road that has a radius of curvature R.

The centripetal force Fc exerted on the moving car is

R

mv

r

mvFc

22

For the car to remain on the curved path, a centripetal force must be provided by the frictional force between the road and the tires.

The car begins to skid on the curve when the centripetal force is greater that the frictional force.

sc fF

μmgR

mv

2max

μgRv max

Force on a Curved PathSafe speed on a curved path may be increased by banking the road along the curve.

In the absence of friction, the reaction force Fn acting on the car must be perpendicular to the road surface.

The vertical component of this force supports the weight of the car.

WθFn cosThat is,

To prevent skidding on a frictionless surface, the total centripetal force must be provided by the horizontal component of Fn;

gR

WvθFn

2

sin

gR

vθ

2

tan

Stable angle of a conic pendulum

v = ?

r

mvθT

2

sin

mgθT cos

gr

vθ

2

tan

Stable angle of a conic pendulum

A Runner on a Curved Track

As the runner rounds the curve, he leans toward the center of rotation.

A Runner on a Curved Track

There are two forces acting on the runner.

W The upward force.

Fcp The centripetal reaction force.

rcp FFW

gR

WvFθF cpr

2

sin WθFr cosgR

vθ

2

tan

The proper angle for a speed of 6.7 m/sec on a 15-m radius is

305.0158.9

)7.6(tan

2

θ 17θ

F = ?如果飛機保持等速

率 ,座椅作用在駕駛員的力量 ,何處最大 ?

如果座椅就是磅秤 ,駕駛員在何處最重 ?

Pendulum

Since the limbs of animals are pivoted at the joint, the swinging motion of animals is basically angular.

Many of the limb movements in walking and running can be analyzed in terms of the swinging movement of a pendulum.

WALKING

RUNNING

Tiger Woods golf swing

Pendulum

Pendulum

g

As the pendulum swings, there is continuous interchange between potential and kinetic energies.

At the extreme of the swing, the pendulum is momentarily stationary.

Here its energy is entirely in the form of potential energy.

FT

At the extreme of the swing

θmgθWFT coscos' θmgθWma sinsin'max

where amax is the maximum tangential acceleration. θga sinmax

Since 2

24

T

lπg

l

Aθ sinand 2

2

max

4

T

Aπa

Pendulum

g

FT

As the pendulum is accelerated to the center, its velocity increases, and the potential energy is converted to kinetic energy.

The pendulum reaches its maximum velocity vmax.

Potential energy at the maximum angle = Kinetic energy at the lowest position

2max2

1)cos1( mvθmgl

)cos1(22max θglv

At small angle θ

2

2

11cos θθ

θl

Aθ sin

22

222max )]

2

11(1[2 A

l

g

l

Aglglθθglv

T

πAA

l

gv

2max

Walking Some aspects of walking can be analyzed in terms of the simple harmonic motion of a pendulum.

The motion of one foot in each step can be considered as approximately a half-cycle of a simple harmonic motion.

A person walks at a rate of 2 steps/sec and that each step is 90 cm long.

In the process of walking each foot rests on the ground for 0.5 sec (T = 1 sec) and then swings forward 180cm and comes to rest again 90 cm ahead of the other foot.

The speed of walking v is m/sec 1.8steps/sec 2cm 90 v

The maximum velocity of the swinging foot vmax is

sec/m 65.51

9028.62max

T

πAv

This is three times faster than the body.

Physical Pendulum

In the discussion of a simple pendulum the total mass of the system is assumed to be located at the end of the pendulum.

A more realistic model of a pendulum should be a physical pendulum, which takes into account that the distribution of mass along the swinging object.

It can be shown that under the force of gravity the period of oscillation T for a physical pendulum is

Wr

IπT 2

Here I is the moment of inertia of the pendulum around the pivot point O. This quantity should be calculated by some integration.

The distance of the center of gravity from the pivot point O is described as “r”.

O

Speed for Walking

Walking and running are two different types of human motions.

In the analysis of walking and running, the leg may be regarded as a physical pendulum.

The moment of inertia I for the leg is

33

22 l

g

WmlI where W is the weight of

the leg and l is its length

If we assume that the center of mass of the leg is at its middle (r = 0.5 l), the period of oscillation is

g

lπ

Wl

lgWπ

Wr

IπT

3

22

2/

)3/)(/(22

2

The period of walking is ~ l1/2

For a 90-cm-long leg, the period is 1.6 second.

Walking

The most effortless walking speed for this person if the length of each step is 90cm is:

s/m125.1)2/6.1/(9.0 v

Speed for Walking

Each step in the act of walking can be regarded as a half-swing of a simple harmonic motion, the number of steps per second is simply the inverse of the half period.

In a most effortless walk, the legs swing at their natural frequency, and the time for one step is T/2.

The speed of walking ∝ Number of steps × The length of the step

lT

v 1

The speed of walking v

Since lT

lll

v 1

Thus, the speed of the natural walk of a person increases as the square root of the length of his legs.

l: the length of the leg

The situation is different when a person (or any animal) runs at full speed.

Whereas in a natural walk the swing torque is produced primarily by gravity, in a fast run the torque is produced mostly by the muscles.

Using some reasonable assumptions, we can show that similarly built animals can run at the same maximum speed, regardless of differences in leg size.

Speed for Running

We assume that the length of the leg muscles is proportional to the length of the leg (l) and that the area of the leg muscles is proportional to l2.

The mass of the leg is proportional to l3.

The maximum force that a muscle can produce (Fm) is proportional to the area of the muscle.

The maximum torque produced buy the muscle is proportional to the product of the force and the length of the leg; that is, 3

max llFm

Speed for Running

max

In general, the period of oscillation for a physical pendulum under the action of a torque with maximum value of is given by

max

2

IT I: the moment of inertia

52mass llI Since

3max llFm

ll

lT

3

5

The maximum speed of running vmax is again proportional to the product of the number of steps per second and the length of the step. Because the length of the step is proportional to the length of the leg, we have

11

1

max ll

lT

v

Speed for Running

max

11

1

max ll

lT

v

This shows that the maximum speed of running is independent of the leg size, which is in accordance with observation.

A fox can run

as fast as a horse.

Exercise:If a person stands on a rotation pedestal with his arms loose, the arms will rise toward a horizontal position. (a) Explain the reason for this phenomenon. (b) Calculate the angular (rotational) velocity of the pedestal for the angle of the arm to be at 60 with respect to the horizontal. What is the corresponding number of revolutions per minute? Assume that the length of the arm is 90 cm and the center of mass is at mid-length.

Answer:

w = 3.546 rad/s = 33.9 rpm