Rotational Vs. translational kinematics

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Intros
Lessons
  1. Rotational Motion
  2. Translational-Rotational Analogues
  3. Tangential Acceleration Vs. Centripetal Acceleration
  4. Rotational Kinematics Equations
  5. Rolling Without Slipping
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Examples
Lessons
  1. A centrifuge starts from rest and has an angular acceleration of 150 rad/s2 for 2.5s.
    1. Find the rotational displacement and number of revolutions per minute.
    2. What is the final angular velocity after 2.5s?
  2. A car engine slows down from 4200rpm to 600rpm in 2.0s.
    1. Calculate the angular acceleration.
    2. Find the total number of revolution the car makes is this time.
  3. A high-speed jet takes 1.5 min to turn through 20 complete revolutions before reaching its final speed.
    1. What is the angular acceleration?
    2. What was the final angular speed in rpm?
  4. To move a large pottery wheel with a radius of 22.0cm, a small rubber wheel is used. Wheels are mounted and their circular edges are in contact. The radius of the small wheel is 3.0cm and it accelerates at the rate of 6.4 rad/s2 without slipping.
    1. Calculate the angular acceleration of the pottery wheel.
    2. How much time does it take for the pottery wheel to reach a speed of 62 rmp?
  5. A car reduces its speed uniformly from 86km/h to 44 km/h, during this time the wheels of the car with a diameter of 0.60m, make 66 revolutions.
    1. Find the angular acceleration of the wheels.
    2. If the car continues to decelerate, how many more revolutions are required to stop the car?
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Topic Notes
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In this lesson, we will learn:

  • Definition of Rotational Motion, Rigid Object and The Axis of Rotation.
  • Translational-Rotational Analogues
  • Tangential Acceleration Vs. Centripetal Acceleration
  • Rotational Kinematics Equations
  • Rolling Without Slipping Motion

Notes:


Rotational Motion
  • To explain the rotational motion, we need to consider a rigid object.

  • A rigid object: it is an object with a definite shape, the particles composing the object are at a fixed position relative to one another.

  • When an object rotates, all the points composing the object move in a circle.

  • Axis of Rotation: The center of all circles lie on a line called Axis of Rotation, which is perpendicular to the page.

Rotational Vs. Translational Kinematic


  • Through rotation, the object changes position and moves through different angles; ΞΈi,ΞΈf \theta _{i}, \theta _{f}


  • Δθ=ΞΈfβˆ’ΞΈi \Delta \theta = \theta_{f} - \theta_{i}

    Δθ\qquad \Delta \theta: Change in position, angular displacement (radians)
    ΞΈf\qquad \theta_{f}: Final position
    ΞΈi\qquad \theta_{i} : Initial positionΒ 

  • Similar to translational motion velocity is defined as a change in position over time elapsed;

  • w=ΔθΔt\large w = \frac{\Delta \theta}{\Delta t}

    w w: angular velocity (radians/sec)

  • Similarly, the acceleration is defined as a change in velocity over time elapsed;

  • βˆβ€‰= \propto \, = Ξ”wΞ”t\large \frac{\Delta w}{\Delta t}

    ∝ \propto: angular acceleration (radians/sec2)

Translational-Rotational Analogues

Rotational Vs. Translational Kinematic


  • Each point of a rotating object has a linear velocity and a linear acceleration, so we have to relate the linear quantities to the angular quantitates.

Rotational Vs. Translational Kinematic


v=v = Ξ”lΞ”t \large \frac{\Delta l}{\Delta t} == rΔθΔt\large \frac{r \Delta \theta }{\Delta t} =rw= rw

v=rw v = rw \, (1)


a=a = Ξ”vΞ”t\large \frac{\Delta v}{\Delta t} == rΞ”wΞ”t\large \frac{r \Delta w}{\Delta t} =r∝ = r \propto

a=rβˆβ€‰a = r \propto \, (2)


Tangential Acceleration Vs. Centripetal Acceleration

In rotational motion, we have to consider the two components of acceleration,

ara_{r} ; tangential component, that changes the magnitude of the velocity and it is tangent to the direction of motion;

ar=r∝a_{r} = r \propto

aca_{c} ; centripetal competent, that changes the direction of velocity and is directed towards the center (caused by centripetal force);

ac=a_{c} = v2r\large \frac{v^{2}}{r}

To find the total acceleration we can find the magnitude of the acceleration using the Pythagoras theorem;

atotal2 = a2T + a2c a^{2}_{total} \, = \, a\frac{2}{T} \, +\, a\frac{2}{c}

atotal = aT2 + ac2a_{total} \, = \, \sqrt{a^{2}_{T} \, + \, a^{2}_{c} }


Rotational Kinematics Equations
The following table compares the rotational kinematics equation to the translational kinematics equation.

Rotational Vs. Translational Kinematic


Conversion Factor
It is very common in rotational motion to measure the angular velocity in round per minutes (rmp), in order to convert to the standard unit of radians per second (rad/s), we use the following the conversion factor;

rmp × 2Ο€β€…β€Šradβ€…β€Š/β€…β€Šrev60β€…β€Šsecβ€…β€Šsecβ€…β€Š/β€…β€Šmin \frac{2 \pi \; rad \; / \; rev}{60\;sec\;sec\; / \; min} rad/s


Rolling Without Slipping
There are many examples in everyday life, where the object rolls but it does not slip.
Examples: rolling ball, rolling wheel.
Rolling without slipping depends on the static friction between the rolling object and the floor.
In this case, we have to consider both rotational and translational motion. To relate these two types of motion, use v=rw v = rw (where rr is the radius).

As the following diagram represents, the wheel is rolling (Rotational Motion) to the right and at the same time, the center of gravity is changing position (Translational Motion).

Rotational Vs. Translational Kinematic

dt\large \frac{d}{t} == rΔθt\large \frac{r \Delta \theta}{t} β‡’v=rw\quad \Rightarrow \quad v = rw


vv : Center of mass speed
rr : RadiusΒ 
ww : Angular speed about the center of massΒ