derive equation of motion for velocity displacement for angular motion.
Answers
What is angular displacement?
Angular displacement is a vector quantity and it the angle subtended by any point of the rotating body with its axis. In S.I. its unit is radian. Angular velocity is the rate of change of angular displacement and it is also a vector quantity.
Derivation
The angular displacement can be calculated by the below formula when the value of initial velocity, acceleration of the object, and time are shared.
θ= wt + 1/2αt^2
Where,
θ- Angular displacement of the object
t- Time
α- angular acceleration
Now, formula for Angular Linear is
In Rotational, kinetic equation is
ω=ω0+αt,
Δθ=ω0t+1/2αt^2,
ω^2=ω02+2αθ,
In translational, kinetic equation is
v=u+at
or s=ut+1/2at^2
v2 = vo2 + 2ax
Where,
ω- Initial angular velocity
Considering an object having a linear motion with initial acceleration a and velocity u, when time t and the final velocity of the object is with the total displacement s then,
a = dv/dt
The change in velocity
The rate which can be written as
dv = a dt
Integrating both the sides, we get,
∫uvdv=a∫0tdt
v – u = at
Also,
a=dv/dt
a=dxdv/dtdx
As we know v=dx/dt, we can write,
a=vdv/dx
v dv=a dx
The equation we get after integrating both sides
∫uvvdv=a∫0sdx
V^2–u^2=2as
From the equation -1 into the equation – 2 by substituting the value of u, we get
V^2−(v−at)^2=2as
2vat–a^2t^2=2as
By dividing the equation of both sides by 2a, we have
s=vt–1/2at^2
And at last, the value of v being substituted by u, we will get.
s=ut+1/2at^2
Answer:
have a look on the attachment
