How Much Angular Displacement Will A Wheel Cover In The Time Between t =5 seconds & t =15 seconds for the given graph?
After completing linear motion kinematics, projectile motion and Newton’s Law of Motion, the next topic which you study in Physics is “Circular Motion”.
Though circular motion and linear motion are much different from each other yet they are interrelated to each other in many aspects.
What Is Circular Motion & Terms Used In It?
Whenever a point mass moves around a fixed point in a circular path, its motion is termed as circulation motion. In linear motion, we study about displacement, velocity, height and acceleration, etc. Whereas, in circular one, we have to study a few extra terms like- angular displacement, angular velocity and angular acceleration etc. but here too, we will need linear ones.
- Angular Displacement :
Angular displacement of a point mass is the angle that measures how many radians it has revolved from its initial position. It is denoted with theta (θ).
- Angular Velocity :
Angular velocity is nothing but the rate of change in angular displacement . It’s measured in radian/sec and denoted with omega (ω).
- Angular Acceleration :
Angular acceleration shows the
rate of change in angular
velocity. It is the second order
derivative of angular
displacement and is denoted by
alpha (α) and measured in
radian/sec^{2}.
- How Are Circular And Linear Motion Related To Each Other?
Circular motion is related to the linear motion in the following ways-
- Relation Between Angular Displacement (θ) And Linear Displacement (s)
In the above diagram, θ is the angular displacemet of a particle moving in a circular path, s is linear displacement of particle and r is the radius of the circular path.
As we know from the basics of mathematics,
angle = arc/ radius ….(i)
From the diagram; angle is θ, radius is r and arc length is s. On substituting this data in equation (i), we get
θ = s/r
or, we can write it s= r θ.
- Relation between angular velocity (ω) and linear velocity (v)
From the relationship between
angular displacement and
linear displacement,
s= r θ …(ii)
differentiating with respect to
time t, we get
ds/dt = d/dt(r θ)
v= r dθ/dt
finally, v= rω …(iii)
In the vector form, it’s written as
v = ω × r where, v, ω and r
in vector form.
- Relation Between Angular Acceleration And Linear Acceleration
From the relationship between angular velocity ω and linear velocity v,
v= rω …(iii)
Now, differentiating this with respect to time,
dv/dt = d/dt(rω)
a = r dω/dt
a = r α …(iv)
In the vector form, it’s written as
a = r × α where, a,α and r
in vector form.
Now, let’s come to the solution of our question.
The equation s= r θ works only in case the data asked in question is instantaneous. Here, in the question, we have to calculate angular displacement between a given time interval. So this approach will not work.
Appropriate Approach To Attempt This Question-
We have to evaluate the instantaneous angular displacement at time t= 15 second and t= 5 second. The difference of θ(t=15) and θ(t=5) will be the resultant angular displacement of the wheel between t= 5 sec and t= 15 sec.
What is the angular displacement of the wheel between t = 5 s and t = 15 s?
Solution:
Instantaneous angular displacement of wheel wheel at t= 5 sec
θ_{1 }= θ(t=5) = 200 rad
And, at t= 15 sec
θ_{2 }= θ(t= 15) = 250 rad
So, resultant angular displacement of wheel between t= 5 sec and t= 15 sec
θ_{resultant }= θ_{2 }– θ_{1}
_{ } = 250 – 200
= 50 radian