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RotationsSo far we have considered translational motion (along some path from point A to B). Things can also rotate. Rotational motion is common. A child's top, a seesaw or a merry-go-round are examples of 'pure' rotational motion.(The motion of a wheel on a car is a mixture of translation and rotational motion.) The seesaw is in translational

equilibrium since the whole seesaw is not moving up, down, left or right. It is also in translational equilibrium since all the forces balance. However the seesaw can rotate about the pivot. If the 'rotational forces' are balanced, it will

not rotate. If they are not balanced e.g. the two weights are not equal and at the same distance from the pivot, the seesaw will rotate. First we will look at doing 'rotational kinematics' in analogue with translational kinematics.

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Angular Displacement, Speed and Acceleration Degrees (o) are not good units for angles for what we want to do. A better unit is the radian. A radian is really dimensionless but I will often write 'rad' to show the number is really an angle measured in radians. So what is a radian? An angle in radians is the arc length you travel around a circle divided by the radius:

Since the circumference of a circle is 2πr there are 2π radians in 360o. That works out to 1 radian = 57.3o. A radian is dimensionless.

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Angular Displacement and Angular Speed

When an object rotates around a fixed axis, it moves through some angle θ. We can thing of θ being the angular position. If the object rotates from θ

0 to θ

1 in

some time Δt the angular speed is: Angular speed = ω = =

θ0 - θ

1 Δθ

ΔtΔt

The units are are rad/s but since radians do not have units it is really s-1 . This is the average angular speed but we can think of a 'snap shot' to give us instantaneous angular speed. Is it a vector? Yes it is a special kind of vector but we will always deal with things rotating in a plane so we will not have to be too concerned with the vector nature of angular speed.

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Angular Acceleration

In an analog to translational motion we can define angular acceleration as change in angular speed, α, for some time interval:

α = ΔtΔω

The units for angular acceleration are s-2.

The sign convention (basically the vector part) for angular displacement, speed and acceleration is that counterclockwise is positive and clockwise is negative.

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Center of Mass and Rotational Inertia If we have a uniform stick hanging from a string placed at the middle of the stick we can spin the stick around. If there is no friction it will spin forever. It takes some 'angular force' (called torque) to get it going. If it is at rest, it will stay at rest. This suggests an angular versions of Newton's 1st laws.

Every extended object (not a point mass) has a place where the object will balance and spin around. This point is called the 'center of mass'. An axis through the center of mass will allow the object to rotate without translational motion.

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Center of mass and Rotational motionAn object can have translational and rotational motion. A high diver's body can rotate about the center of mass as well as move along a parabolic path as we discussed in the last chapter. Consider tossing a bowling pin:

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Rotational Inertia

The resistance of an object to start rotating or stop rotating i.e., change its angular speed, is rotational inertia. The measure of this rotational mass is called the moment of inertia. This property of an extended object depends on the mass of the object and the shape (geometry) of the object. The units for this rotational mass are kg m2

.

The rotational mass also depends on which axis the object is rotating about. For a uniform sphere it does not matter which axis as long as it goes through the center. For a long rod, the rotational mass is different through the long axis through the center of mass than perpendicular through the center.

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Torque

Torque is rotational force. We can define torque more precisely using the analogue of Newton's 2nd Law for rotations.

Torque = rotational mass x angular acceleration

or

τ = I α (Newton's 2nd law for rotation)

We know from a seesaw that a larger child need to be positioned closer to the pivot than a smaller child for the seesaw to balance. A lever can be used to lift a heavy object with a smaller force.

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TorqueMechanical advantage of a lever is contained in the definition of torque:

Torque = (magnitude of Force) x (lever arm)

The lever arm is the perpendicular distance from the 'line of action' of the force and the axis of rotation. The line of action is an imaginary line collinear with the force. The units are Nm (not joules!). Torque is positive if it produces a counter- clockwise rotation and negative if the rotation is clockwise

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Friction

So far we have ignored friction. Friction can be good or bad. We could not walk without friction but eliminating friction in machinery is very important i.e. using lubricants.

If you push something across the floor, you may notice that it is harder (requires more force) to get it going than it is to keep it going. There are two types of friction: static and kinetic. Static friction is stronger than kinetic friction.

Friction of object sliding over one another is causes by small microscopic bumps. Make the surfaces flatter and the friction is reduced. Add oil and the friction is even smaller.

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Friction

A rolling wheel is actually a case of static friction. The wheel is moving and rotating. However as the wheel rotates around, the wheel comes down on the ground and then lifts off the ground. There is no motion between the ground and wheel at the point of contact at the bottom of the wheel. This assumes there is no slipping.

ABS braking systems on cars are designed to keep the wheel from slipping by actually removing the brakes for a short time. Kinetic friction is very bad when you want to Stop quickly.

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Work and Energy

In the last chapter, we discussed work. Work is defined as force times distance and is a scalar (simple number).

Work = 'Force' x distance = F d

The units are joules.

We also discuss gravitational potential (stored) energy. Gravitational potential energy is:

Grav. potential energy = GPE = mgh = weight x height

The units are the same as the units for work … joules or kg m2/s2.

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Kinetic EnergyA more obvious form of energy is energy of motion or kinetic energy. The energy of an object due to its motion is:

Kinetic Energy = KE = ½ mass x speed2 = ½mv2

The unit is again a joule. Note that if you double the speed of an object, the kinetic energy is four times larger because of the squared speed.

An object can also have kinetic energy because it is rotating. Then:

Rotational KE = ½ I 2

Where I is the rotational 'mass' and ω is the angular speed. Note it is just like translational KE except it has rotational variable.

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Work Energy TheoremWork is something done to an object. Energy is something an object has. Said another way, when you do work on an object, you give it energy. The work energy theorem says the work you do on something is equal to the change in theKinetic energy:

W = ∆KE

The caveat to this statement is the potential energy does not change like going up or down a hill. Friction (which is turned into heat) is another caveat so the work energy theorem also assumes there is no friction.

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Work Energy Theorem

If you push your friends stalled car with a force of 500 N for a distance of 1 m, you do work on the car.

W = F d = 500 N . 1 m = 500 J

Assuming no friction and a level road:

W = 500 J = KE = ½ mv2

If the mass of the car is 1000 kg then the final speed of the car is:

v2 = (2 . 500 J)/1000 kg = 1 m2/s2

v = 1 m/s

Note this is actually the change in velocity since it is ∆KE.

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Conservation of Energy

The work energy theorem is just a special case of the more general law of energy conservation:

Energy cannot be created or destroyed. Energy can be changed from one form to another but the total amount never changes.

Energy conservation is probably the single most important idea in physics. When we say energy we are now including kinetic energy as well as all forms of potential energy (and eventually we will add heat energy).

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PowerThe rate work is done or the time rate of change of energy is called power.

Power =

The unit of power is the J/s or Watt.

If you use a 20 W electrical clock radio you are using (and paying) for energy. A 1 kW (1000 W) hair dryer used 50 times the energy as the radio each second.

If you get an electrical bill, you may have noticed you are charged by the kw.hr. A kw.hr is one kilowatt for 1 hour or 3.6 x 106 J (3.6 MJ).

Work done

Time interval

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Momentum

Until the 1850s, scientist argued about the most important quantity of motion they called the 'Impedo’. It was one of those arguments where both sides were correct. One group argued for what we now call energy conservation and one group for what we now call momentum conservation.

Momentum is:

momentum = mass . velocity = mv

The units are kg m/s (no special name). Momentum is a vector because of the velocity term. A big mass like a ship can give something a big momentum. A large velocity can also give something a large momentum like a bullet.

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ImpulseIt requires a force acting for some time to change the velocity and thus the momentum of an object. A force acting on something for some time interval is called 'impulse' and is related the the change in momentum.

Impulse = force x time interval = F ∆t = ∆mv

This is really just Newton's 2nd law:

a = =

F ∆t = ∆mv

A bullet hitting a brick wall has a large impulse because the time interval is small. A supertanker need a small force for hours to change its momentum.

Fm

change in vchange in time

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Conservation of MomentumWe two objects collide the total momentum is conserved:

total moment before = total momentum after

This is known as the law of momentum conservation. The caveat on this is that there are no external forces (internal between the two object is fine.)

Momentum is a vector so we have to have the same amount of momentum before and after in each dimension e.g. x and y.

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Conservation of Angular Momentum

Our third classical conservation law is 'Conservation of Angular Momentum'. Angular momentum, L, is just rotational momentum:

L = I .ω

where I is the rotational mass (moment of inertia) and ω is the angular velocity. Note the analogy with linear momentum.It is a vector but we will stay in a plane like with rotational motion so we can ignore the vector nature (but not the sign on quantities). The re-done form of Newton's 2nd law also applies:

∆L = τ . t

'τ.t' is called the 'angular impulse'.

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Conservation of Angular MomentumThink about an ice skater spinning around. If there is no friction with the ice, angular momentum will be conserved and the skater will continue to spin. If the skater pulls in her arms, this will lower the angular mass (I). To keep the angular moment the same, the angular momentum (ω) will have to increase and she will spin faster.

If the skater digs her skate into the ice, an external torque (τ) is applied for some time (t) giving an angular impulse which will reduce the total angular momentum (L) and angular speed (ω). She then slows down or stops.

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Conservation Laws

We have introduced the three 'classical' conservation laws:Each conservation law is based on a symmetry. The three conservation laws and associated symmetries are:

Energy Conservation – Time reversal symmetryMomentum Conservation – Translational symmetryAngular Momentum Conservation - Rotational symmetry

A symmetry is something you can 'do' to a physical system which does not change the laws of physics. Nearly all of our understanding of how the physical world works can be traced back to a symmetry principle.

(Note: this is not in the text book).