Answer
Verified
429.6k+ views
Hint:The angle in radians through which a point or line has been rotated about an axis is called its angular position. It is a vector quantity. Angular velocity is defined as how fast the object rotates or changes its position with time. It is represented by using the symbol omega $\omega $ .
Complete step-by-step answer:
Step I:
Given that
angular velocity$\omega $ varies with time according to equation, $\omega = at + b$ ---(i)
When t = $0$, angular velocity $\omega = 0.1rad/s$
Angular position $\theta = 2\pi rad$
Substituting the values in equation(i),
$0.1 = a \times 0 + b$
$b = 0.1$
Substitute value of ‘b’ in equation (i),
$\omega = at + 0.1$ ---(ii)
Step II:
Angular velocity is given by
$\omega = \dfrac{{d\theta }}{{dt}}$
$d\theta = \omega .dt$
$\int {d\theta = \int {\omega .dt} } $
Integrating the above equation,
$\int\limits_0^\theta {d\theta = \int\limits_0^t {(at + 0.1)dt} } $
\[[\theta ]_0^2 = \dfrac{{a{t^2}}}{2} + 0.1t + 2\]
Step III:
Also given when $t = 2\sec $
$\omega = 5rad/s$
Substituting these values in equation(ii)
$5 = 2a + 0.1$
$2a = 5 - 0.1$
$a = \dfrac{{4.9}}{2}$
$a = 2.45$
Step IV:
Substituting the value of a in equation (ii)
$\omega = 2.45t + 0.1$
Angular acceleration is the rate of change of the angular velocity and it is a vector quantity. It is given by, $ \propto = \dfrac{{d\omega }}{{dt}}$
Substituting value of $\omega $ and solving for angular acceleration
$ \propto = \dfrac{{d\{ 2.45t + 0.1\} }}{{dt}}$
$ \propto = 2.45rad/{s^2}$
Step V:
Angular position at $t = 4s$
$\theta = (2.45) \times \dfrac{{{{(4)}^2}}}{2} + 0.1 \times 4 + 2$
$\theta = 19.6 + 0.4 + 2$
$\theta = 22rad$
Step VI:
Therefore, the when $t = 4s$
Angular position of the disc is $\theta = 22rad$ and
Angular acceleration of the disc is $ \propto = 2.45rad/{s^2}$
Note:Sometimes there can be confusion between angular frequency and velocity. It is important to note that angular velocity and angular frequency are the same terms. Angular frequency is the magnitude of the angular velocity. It is therefore sometimes also known as the angular velocity. Angular velocity is the product of frequency and the constant $2\pi $. Whenever any object makes one complete revolution in one second, the object is said to have rotated to a measure of $2\pi $ radians per second.
Complete step-by-step answer:
Step I:
Given that
angular velocity$\omega $ varies with time according to equation, $\omega = at + b$ ---(i)
When t = $0$, angular velocity $\omega = 0.1rad/s$
Angular position $\theta = 2\pi rad$
Substituting the values in equation(i),
$0.1 = a \times 0 + b$
$b = 0.1$
Substitute value of ‘b’ in equation (i),
$\omega = at + 0.1$ ---(ii)
Step II:
Angular velocity is given by
$\omega = \dfrac{{d\theta }}{{dt}}$
$d\theta = \omega .dt$
$\int {d\theta = \int {\omega .dt} } $
Integrating the above equation,
$\int\limits_0^\theta {d\theta = \int\limits_0^t {(at + 0.1)dt} } $
\[[\theta ]_0^2 = \dfrac{{a{t^2}}}{2} + 0.1t + 2\]
Step III:
Also given when $t = 2\sec $
$\omega = 5rad/s$
Substituting these values in equation(ii)
$5 = 2a + 0.1$
$2a = 5 - 0.1$
$a = \dfrac{{4.9}}{2}$
$a = 2.45$
Step IV:
Substituting the value of a in equation (ii)
$\omega = 2.45t + 0.1$
Angular acceleration is the rate of change of the angular velocity and it is a vector quantity. It is given by, $ \propto = \dfrac{{d\omega }}{{dt}}$
Substituting value of $\omega $ and solving for angular acceleration
$ \propto = \dfrac{{d\{ 2.45t + 0.1\} }}{{dt}}$
$ \propto = 2.45rad/{s^2}$
Step V:
Angular position at $t = 4s$
$\theta = (2.45) \times \dfrac{{{{(4)}^2}}}{2} + 0.1 \times 4 + 2$
$\theta = 19.6 + 0.4 + 2$
$\theta = 22rad$
Step VI:
Therefore, the when $t = 4s$
Angular position of the disc is $\theta = 22rad$ and
Angular acceleration of the disc is $ \propto = 2.45rad/{s^2}$
Note:Sometimes there can be confusion between angular frequency and velocity. It is important to note that angular velocity and angular frequency are the same terms. Angular frequency is the magnitude of the angular velocity. It is therefore sometimes also known as the angular velocity. Angular velocity is the product of frequency and the constant $2\pi $. Whenever any object makes one complete revolution in one second, the object is said to have rotated to a measure of $2\pi $ radians per second.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
10 examples of evaporation in daily life with explanations
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Why is there a time difference of about 5 hours between class 10 social science CBSE