The angular velocity is a measure of the speed of rotation and is defined as the angle that the position vector of the rotating object rotates, per unit of time. It is a magnitude that describes very well the movement of a multitude of objects that constantly rotate everywhere: CDs, car wheels, machinery, the Earth and many more..
A diagram of the "London eye" can be seen in the following figure. It represents the movement of a passenger represented by point P, which follows the circular path, called c:
The passenger occupies position P at instant t and the angular position corresponding to that instant is ϕ.
From the instant t, a period of time Δt elapses. In that period the new position of the punctual passenger is P 'and the angular position has increased by an angle Δϕ.
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For rotational quantities, Greek letters are widely used in order to differentiate them from linear quantities. So initially we define the mean angular velocity ωm as the angle traveled in a given period of time.
Then the quotient Δϕ / Δt will represent the mean angular velocity ωm between times t and t + Δt.
If you want to calculate the angular velocity just at the instant t, then we will have to calculate the quotient Δϕ / Δt when Δt ➡0:
Linear speed v, is the quotient between the distance traveled and the time taken to travel it.
In the figure above, the arc traveled is Δs. But that arc is proportional to the angle traveled and the radius, the following relationship being fulfilled, which is valid as long as Δϕ is measured in radians:
Δs = r ・ Δϕ
If we divide the previous expression by the time lapse Δt and take the limit when Δt ➡0, we will obtain:
v = r ・ ω
A rotational movement is uniform if at any observed instant, the angle traveled is the same in the same period of time.
If the rotation is uniform, then the angular velocity at any instant coincides with the mean angular velocity.
Furthermore, when a complete turn is made, the angle traveled is 2π (equivalent to 360º). Therefore in a uniform rotation the angular velocity velocidad is related to the period T, by the following formula:
f = 1 / T
That is, in a uniform rotation, the angular velocity is related to the frequency by:
ω = 2π ・ f
The cabs of the great spinning wheel known as the "London eye”They move slowly. The speed of the cabs is 26 cm / s and the wheel is 135 m in diameter.
With these data calculate:
i) The angular velocity of the wheel
ii) The frequency of rotation
iii) The time it takes for a cabin to make a complete turn.
Answers:
i) The speed v in m / s is: v = 26 cm / s = 0.26 m / s.
The radius is half the diameter: r = (135 m) / 2 = 67.5 m
v = r ・ ω => ω = v / r = (0.26 m / s) / (67.5 m) = 0.00385 rad / s
ii) ω = 2π ・ f => f = ω / 2π = (0.00385 rad / s) / (2π rad) = 6.13 x 10-4 turns / s
f = 6.13 x 10 ^ -4 turn / s = 0.0368 turn / min = 2.21 turn / hour.
iii) T = 1 / f = 1 / 2.21 lap / hour = 0.45311 hour = 27 min 11 sec
A toy car moves on a circular track with a radius of 2m. At 0 s its angular position is 0 rad, but after a time t its angular position is given by:
φ (t) = 2 ・ t
Determine:
i) The angular velocity
ii) The linear speed at any instant.
Answers:
i) The angular velocity is the derivative of the angular position: ω = φ '(t) = 2.
That is, the toy car at all times has constant angular velocity equal to 2 rad / s.
ii) The linear speed of the car is: v = r ・ ω = 2 m ・ 2 rad / s = 4 m / s = 14.4 Km / h
The same car from the previous exercise begins to stop. Its angular position as a function of time is given by the following expression:
φ (t) = 2 ・ t - 0.5 ・ ttwo
Determine:
i) The angular velocity at any instant
ii) The linear speed at any instant
iii) The time it takes to stop from the moment it begins to decelerate
iv) The angle traveled
v) distance traveled
Answers:
i) The angular velocity is the derivative of the angular position: ω = φ '(t)
ω (t) = φ '(t) = (2 ・ t - 0.5 ・ ttwo) '= 2 - t
ii) The linear speed of the car at any instant is given by:
v (t) = r ・ ω (t) = 2 ・ (2 - t) = 4 - 2 t
iii) The time it takes to stop from the moment it begins to decelerate, is determined by knowing the moment in which the velocity v (t) becomes zero.
v (t) = 4 - 2 t = 0 => t = 2
That is, it stops 2 s after starting to brake.
iv) In the period of 2s from the moment it starts to brake until it stops, an angle given by φ (2) is traveled:
φ (2) = 2 ・ 2 - 0.5 ・ 2 ^ 2 = 4 - 2 = 2 rad = 2 x 180 / π = 114.6 degrees
v) In the period of 2 s from when it starts to brake until it stops, a distance s given by:
s = r ・ φ = 2m ・ 2 rad = 4 m
The wheels of a car are 80 cm in diameter. If the car travels at 100 km / h. Find: i) the angular speed of rotation of the wheels, ii) the frequency of rotation of the wheels, iii) The number of turns the wheel makes in a journey of 1 hour.
Answers:
i) First we are going to convert the speed of the car from Km / h to m / s
v = 100 Km / h = (100 / 3.6) m / s = 27.78 m / s
The angular speed of rotation of the wheels is given by:
ω = v / r = (27.78 m / s) / (0.4 m) = 69.44 rad / s
ii) The frequency of rotation of the wheels is given by:
f = ω / 2π = (69.44 rad / s) / (2π rad) = 11.05 turn / s
The rotation frequency is usually expressed in revolutions per minute r.p.m.
f = 11.05 turn / s = 11.05 turn / (1/60) min = 663.15 r.p.m
iii) The number of laps the wheel makes in a 1 hour journey is calculated knowing that 1 hour = 60 min and that the frequency is the number of laps N divided by the time in which those N laps are made.
f = N / t => N = f ・ t = 663.15 (turns / min) x 60 min = 39788.7 turns.
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