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angular motion - Coggle Diagram
angular motion
angular displacement
definition:
the angle and the direction through which a body turns
how is it measured?
1 revolution = 360 degrees = 2π radians
radian (rad)
revolution
denoted by θ
pure number (no dimensions)
angular velocity
definition:
the rate of change of angular velocity with respect to time
how is it measured?
radian per second (rad/s)
revolution per minute (rpm)
denoted by ω
formula:
ω = 𝑑θ/𝑑𝑡
ω = angular velocity [rad/s]
θ = angular displacement [rad]
t = time [s]
angular acceleration
definition:
the rate of change of angular velocity with respect to time
how is measured?
radian per second (rad/s^2)
denoted by α
formula:
α = 𝑑ω/𝑑𝑡 or 𝑑^2θ/𝑑𝑡^2
α = angular acceleration [rad/s^2]
ω = angular velocity [rad/s]
θ = angular displacement [rad]
t = time [s]
rotational motion
formula:
ωf = ωi + α t
ωf^2 = ωi^2 + 2 a (θf - θi)
θf = θi + ωi t + 1/2 α t^2
θf - θi= θi + 1/2(ωf + ωi)(t)
where
ωi = initial angular velocity [rad/s]
ωf = final angular velocity [rad/s]
α = angular acceleration [rad/s2]
θi = initial angular displacement [rad]
θf = final angular displacement [rad]
t = time [s]
linear motion
formula:
v = u + a t
sf = si + u t + 1/2 a t^2
v^2 = u^2 + 2 a (sf - si)
sf - si = 1/2 (v + u)(t)
where
v = final velocity [m/s]
u = initial velocity [m/s]
a = acceleration [m/s2
sf = final position [m]
si = initial position
relationship between linear and
rotational motion
formula:
s = r θ
v = r ω
at = r α
an = r ω^2 = 𝑣^2/r
where
r = radius of rotation [m]
s = linear displacement [m]
θ = angular displacement [rad]
v = tangential velocity [m/s]
ω = angular velocity [rad/s]
at = tangential acceleration [m/s^2]
= angular acceleration [rad/s^2]
an = normal or centripetal acceleration [m/s^2]
rolling motion = rotational motion + translational motion
translational motion
every particle in the rigid body has the same instantaneous velocity (no rotation)
rotational motion
every particle in the rigid body has the same angular velocity and travels in circles around a fixed axis
normal or centripetal acceleration
an = 𝑣^2/r or r ω^2
atotal^2 = at^2 + an^2
an = normal acceleration
v = linear velocity
r = radius
ω = angular velocity
atotal = total acceleration
at = tangential acceleration
