# 6.3 Circular motion and rotational kinematics (application)

 Page 1 / 1
Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

## Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the circular motion and rotational kinematics. The questions are categorized in terms of the characterizing features of the subject matter :

• Measurement of angular displacement
• Angular speeds
• Centripetal acceleration

## Measurement of angular displacement

Problem : A particle in uniform circular motion about the center has angular velocity "ω". What is its angular velocity with respect to a point "P" on the circumference of the circle ?

Solution : Angular velocity is a measure of angle in unit time. In the question, measurement of angular velocity about the center of circle is given. It is, therefore, imperative that we seek a relation of angles formed by the motion of the particle at two points of references.

We consider a small arc AA' as shown in the figure, which is covered by the particle in time "dt". By geometry, if the arc subtends an angle "dθ" at "P", then the arc subtends an angle "2dθ" at the center.

Let ${\omega }_{P}$ be the angular velocity of the particle with respect to point "P", then

$\begin{array}{l}{\omega }_{P}=\frac{đ\theta }{đt}\end{array}$

From the relation between angles as obtained earlier, the angular velocity of the particle with respect to center is :

$\begin{array}{l}⇒\omega =\frac{đ\left(2\theta \right)}{đt}=2\frac{đ\theta }{đt}=2{\omega }_{P}\end{array}$

$\begin{array}{l}⇒{\omega }_{P}=\frac{\omega }{2}\end{array}$

## Angular speeds

Problem : Let ${\omega }_{H}$ and ${\omega }_{E}$ respectively be the angular speeds of the hour hand of a watch and that of the earth around its own axis. Compare the angular speeds of earth and hour hand of a watch.

Solution : The time periods of hour hand and that of the earth are 12 hours and 24 hours respectively. Now,

$\begin{array}{l}{\omega }_{H}=\frac{2\pi }{12}\end{array}$

and

$\begin{array}{l}{\omega }_{E}=\frac{2\pi }{24}\\ ⇒{\omega }_{H}=2{\omega }_{E}\end{array}$

Problem : A particle is kept on a uniformly rotating turn table at a radial distance of 2 m from the axis of rotation. The speed of the particle is “v”. The particle is then shifted to a radial distance of 1 m from the axis of rotation. Find the speed of the particle at the new position.

Solution : The angular speed of rotation is constant. Now, linear velocities of the particle at the two positions are :

$\begin{array}{l}{v}_{1}=\omega {r}_{1}=\omega x2=v\phantom{\rule{2pt}{0ex}}\mathrm{\left( given\right)}\\ {v}_{2}=\omega {r}_{2}=\omega x1=\omega \\ ⇒{v}_{2}=\frac{v}{2}\end{array}$

## Centripetal acceleration

Problem : Find the centripetal acceleration (in $\phantom{\rule{2pt}{0ex}}\mathrm{km}/{\mathrm{hr}}^{2}$ ) at a point on the equator of the earth (consider earth as a sphere of radius = 6400 km).

Solution : The centripetal acceleration in terms of angular speed is given by :

$\begin{array}{l}a={\omega }^{2}r\end{array}$

Now, angular speed, in terms of time period is :

$\begin{array}{l}\omega =\frac{2\pi }{T}\end{array}$

Combining two equations, we have :

$\begin{array}{l}a=\frac{4{\pi }^{2}r}{{T}^{2}}\end{array}$

$\begin{array}{l}⇒a=\frac{4x{\left(3.14\right)}^{2}x6400}{{24}^{2}}=439\phantom{\rule{2pt}{0ex}}\mathrm{km}/{\mathrm{hr}}^{2}\end{array}$

#### Questions & Answers

A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
water boil at 100 and why
what is upper limit of speed
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
which colour has the shortest wavelength in the white light spectrum
how do we add
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
what is energy?
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
n=a+b/T² find the linear express
أوك
عباس
Quiklyyy
Moment of inertia of a bar in terms of perpendicular axis theorem
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate