ANGULAR POSITIONdanny/4_rot.pdf4. Rotational Kinematics and Dynamics 1 ANGULAR POSITION • To describe rotational motion, we define angular quantities that are analogous to linear

4. Rotational Kinematics and Dynamics

1

ANGULAR POSITION

• To describe rotational motion, we define angular

quantities that are analogous to linear quantities

• Consider a bicycle wheel that is free to rotate about its

axle

• The axle is the axis of rotation for the wheel

• If there is a small spot of red paint on the tire, we can

use this reference to describe its rotational motion

• The angular position of the spot is the angle θ, that a

line from the axle to the spot makes with a reference

line

• SI unit is the radian (rad)

• θ > 0 – anticlockwise rotation: θ < 0 – clockwise rotation

• A radian is the angle for which the arc length, s, on a

circle of radius r is equal to the radius of the circle

• The arc length s for an arbitrary angle θ measured in

radians is s = r θ

• 1 revolution is 360°= 2π rad

• 1 rad = 360°/2π = 57.3°


2

ANGULAR VELOCITY

• As the bicycle wheel rotates, the angular position of the

spot changes

• Angular displacement is ∆θ = θf – θi

• Average angular velocity is ωav = ∆θ/∆t (rad/s)

• Analogous average linear velocity vav = ∆x/∆t

• Instantaneous angular velocity is the limit of ωav as

the time interval ∆t reaches zero

• ω > 0 – anticlockwise rotation: ω < 0 clockwise rotation

• The time to complete one revolution is known as the

period, T

• T = 2π/ω seconds


3

ANGULAR ACCELERATION

• If the angular velocity of the rotating bicycle wheel

increases or decreases with time, the wheel

experiences an angular acceleration, α

• The average angular acceleration is the change in

angular velocity in a given time interval

• αav = ∆ω/∆t rad/s2

• The instantaneous angular acceleration is the limit of

αav as the time interval ∆t approaches zero

• The sign of angular acceleration is determined by

whether the change in angular velocity is positive or

negative

• If ω is becoming more positive

(ωf > ωi), α is positive

• If ω is becoming more negative

(ωf < ωi), α is negative

• If ω and α have the same sign,

speed of rotation increasing

• If ω and α have opposite signs,

speed of rotation decreasing


4

ROTATIONAL KINEMATICS

• Rotational kinematics describes rotational motion

• Consider the pulley shown below, which has a string

wrapped around its circumference with a mass

attached to its free end

• When the mass is released, the pulley begins to rotate

– slowly at first, then faster and faster

• The pulley thus accelerates with constant angular

acceleration: α = ∆ω/∆t

• If the pulley starts with initial angular velocity ω0 at time

t = 0, and at the later time t the angular velocity is ω

then α = ∆ω/∆t = (ω – ω0)/(t – t0) = (ω – ω0)/t

• Thus the angular velocity ω varies with time as follows:

ω = ω0 + αt

• Example: If the

angular velocity of the

pulley is -8.4rad/s at a

given time, and its

angular acceleration is

-2.8rad/s2, what is the

angular velocity of the

pulley 1.5s later?


5

LINEAR AND ANGULAR ANALOGIES


6

ROTATIONAL KINEMATICS:

EXAMPLE (1)

• To throw a curve ball, a baseball pitcher gives the ball

an initial speed of 36.0 rad/s. When the catcher gloves

the ball 0.595s later, its angular speed has decreased

(due to air resistance) to 34.2 rad/s. What is the ball’s

angular acceleration, assuming it to be constant? How

many revolutions does the ball make before being

caught?


7

ROTATIONAL KINEMATICS:

EXAMPLE (2)

• On a TV game show, contestants spin a wheel when it

is their turn. One contestant gives the wheel an initial

angular speed of 3.4 rad/s. It then rotates through1 ¼

revolutions and comes to rest on the BANKRUPT

space. Find the angular acceleration of the wheel,

assuming it to be constant. How long does it take for

the wheel to come to a rest?


8

TANGENTIAL SPEED OF A

ROTATING OBJECT

• Consider somebody riding a merry-go-round, which completes one circuit every T = 7.5s

• Thus ω = 2π/T = 0.838 rad/s

• The path followed is circular, with the centre of the circle at the axis of rotation

• The rider is moving in a direction that is tangential to the circular path

• The tangential speed is the speed at a tangent to the circular path, and is found by dividing the circumference by T: vt = 2πr/T m/s

• Because 2π/T = ω we have: vt = rω m/s

• Example: Find the angular speed a CD must have to give a linear speed of 1.25m/s when the laser beam shines on the disk 2.50cm and 6.00cm from its centre


9

CENTRIPETAL ACCELERATION OF

A ROTATING OBJECT

• When an object moves in a circular path, it experiences

a centripetal acceleration, acp, which is always directed

toward the axis of rotation

• acp = v2/r

• However v = vt = rω, so acp = (rω)2/r = rω2 m/s2

• Rotating devices known as centrifuges can produce

centripetal accelerations many times greater than

gravity, such as those used to train astronauts, or

microhematocrit centrifuges used to separate blood

cells from plasma

• Example: In a microhematocrit centrifuge, small

samples of blood are placed in capillary tubes. These

tubes are rotated at 11,500rpm, with the bottom of the

tubes 9.07cm from the axis of rotation. Find the linear

speed of the bottom of the tubes. What is the

centripetal acceleration at the bottom of the tubes?


10

TANGENTIAL AND CENTRIPETAL

ACCELERATION

• When the angular speed of an object in a circular path

changes, so does its tangential speed

• When tangential speed changes, a tangential

acceleration is experienced at

• If ω changes by the amount ∆ω, with r remaining

constant, the corresponding change in tangential speed

is ∆vt = r∆ω

• If ∆ω occurs in time interval ∆t, then the tangential

acceleration is at = ∆vt/∆t = r∆ω/∆t

• Since ∆ω/∆t is the angular acceleration α, then the

tangential acceleration of a rotating object is given

by at = rα m/s2

• Recall that at is due to a changing tangential speed,

and that acp is caused by a changing direction of motion

(even if at remains constant)

• In cases where both tangential and centripetal

accelerations are present, the total sum is the vector

sum of the two

• and are at right angles, hence the magnitude of

the total acceleration is a = √(at2 + acp

2)

• The direction is given by φ = tan-1(acp/at)

tar

cpar


11

TANGENTIAL AND CENTRIPETAL

ACCELERATION: EXAMPLE

• Suppose the centrifuge above is starting up with a

constant angular acceleration of 95.0 rad/s2. What is

the magnitude of the centripetal, tangential and total

accelerations of the bottom of a tube when the angular

speed is 8.00 rad/s? What angle does the total

acceleration make with the direction of motion?


12

TORQUE: WHEN FORCE APPLIED IS

TANGENTIAL

• Trying to loosen a nut by rotating a wrench

anticlockwise is easier when you apply the force as far

away from the nut as possible

• Likewise to open a revolving door is easier when you

push further from the axis of rotation

• The tendency for a force to cause a rotation increases

with the distance r from the axis of rotation to the force

• Torque is a quantity that takes into account both the

magnitude of the force and the distance from the axis of

rotation, r

• Torque: τ = rF Nm (Newton – metre)

• This equation is only valid when the applied force is

tangential to a circle of radius r centred on the axis of

rotation


13

TORQUE: WHEN FORCE APPLIED IS

NOT TANGENTIAL

• Consider pulling on a merry-go-round in a direction that is radial (along a line that extends through the axis of rotation)

• Such a force has no tendency to cause a rotation, and thus the axle simply exerts an equal and opposite force, and thus the merry-go-round remains at rest

• A radial force produces zero torque

• If the force applied is at an angle θ to the radial line, the vector force needs to be resolved into radial and tangential components

• Radial component magnitude: Fcosθ

• Tangential component magnitude: Fsinθ

• Only tangential component causes rotation, thus Fcosθ= 0

• General definition of torque: τ = rFsinθ Nm

• τ > 0 – anticlockwise angular acceleration

• τ < 0 – clockwise angular acceleration

Fr


14

TORQUE: EXAMPLE

• Two helmsmen, in disagreement about which way to turn a ship, exert different forces on the ship’s wheel. The wheel has a radius of 0.74m, and the two forces have the magnitudes F1 = 72N and F2 = 58N. Find the torque caused by and the torque caused by . In which direction does the wheel turn as a result of these two forces.

1Fr

2Fr


15

TORQUE AND ANGULAR

ACCELERATION

• A single torque, τ, acting on an object causes the object

to have an angular acceleration α

• Consider a small object of mass m connected to an

axis of rotation by a light rod of length r

• If a tangential force of magnitude F is applied to the

mass, it will move with an acceleration according to

Newton’s 2nd law, a = F/m

• Linear and angular accelerations related by α = a/r

• Combining: α = a/r = F/mr

• Multiplying by r/r gives α = rF/mr2

• Since torque τ = rF, we define a new quantity called the

moment of inertia: I = mr2

• Thus α = τ/I or τ = Iα

• In a system with more than one torque, we take the net

sum of all the torques acting: τnet = Στ = Iα

• Above is Newton’s 2nd law

for rotational motion


16

MOMENT OF INERTIA• I = mr2 is general case for moment of inertia


17

TORQUE AND ANGULAR

ACCELERATION: EXAMPLES

• A light rope wrapped around a disk shaped pulley is

pulled tangentially with a force of 0.53N. Find the

angular acceleration of the pulley, given that its mass is

1.3kg and its radius is 0.11m.

• A fisherman is dozing when a fish takes the line and

pulls it with a tension T. The spool of the fishing reel is

at rest initially and rotates without friction as the fish

pulls for time t. If the radius of the spool is R and its

moment of inertia is I, find the angular displacement of

the spool. Also find the length of line pulled from the

spool and the angular speed of the spool. Hint: make

use of θ = θ0 + ω0t + ½ αt2 and ω = ω0 + αt

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ANGULAR POSITIONdanny/4_rot.pdf4. Rotational Kinematics and Dynamics 1 ANGULAR POSITION • To describe rotational motion, we define angular quantities that are analogous to linear