2.2 - Motion in a circle
● Circular motion - always measured in rad
● ***Instantaneous velocity: in circular motion, speed remains constant BUT velocity
changes constantly, due to change in direction
○ *** v (velocity [ms−1 ])=ω (angular velocity , average [rad s−1 ]) ×r (radius[ m])
■ ***Angular velocity: a pseudovector representation of how fast the
angular position/orientation of an object changes in respect to time
● ***
−1 Δ θ(angle travelled [rad ])
ω (angular velocity , average [rad s ])=
Δ t(time [s])
r∆θ s 2π
● ***ω r = = = v; T =
∆t ∆t ω
2
2 r θ
● *** A( area of sector ,covered during travel [m ])=
2
● Derivation of the equation for centripetal acceleration
s( arc length[ m])=∆ θ(angle travelled [rad ])×r (radius[m])
−1
s Δv (instantaneous velocity , change∈[ m s ])
=∆ θ= {using similar triangles}
r −1
v (instantaneous velocity [m s ])
s
v= , s=v ∆ t
∆ t(time [s])
v ∆t ∆v r∆v 2 v2 ∆ v
= , v ∆t= , v ∆ t=r ∆ v , = =a(acceleration[m s−2 ])
r v v r ∆t
v2 m v2
∵ F=ma , a= , ∴ F= /¿ (where F usually refers to T [tension])
r r
2
v 2
○ ***a= , F=mr ω
r
1
● Circular motion - always measured in rad
● ***Instantaneous velocity: in circular motion, speed remains constant BUT velocity
changes constantly, due to change in direction
○ *** v (velocity [ms−1 ])=ω (angular velocity , average [rad s−1 ]) ×r (radius[ m])
■ ***Angular velocity: a pseudovector representation of how fast the
angular position/orientation of an object changes in respect to time
● ***
−1 Δ θ(angle travelled [rad ])
ω (angular velocity , average [rad s ])=
Δ t(time [s])
r∆θ s 2π
● ***ω r = = = v; T =
∆t ∆t ω
2
2 r θ
● *** A( area of sector ,covered during travel [m ])=
2
● Derivation of the equation for centripetal acceleration
s( arc length[ m])=∆ θ(angle travelled [rad ])×r (radius[m])
−1
s Δv (instantaneous velocity , change∈[ m s ])
=∆ θ= {using similar triangles}
r −1
v (instantaneous velocity [m s ])
s
v= , s=v ∆ t
∆ t(time [s])
v ∆t ∆v r∆v 2 v2 ∆ v
= , v ∆t= , v ∆ t=r ∆ v , = =a(acceleration[m s−2 ])
r v v r ∆t
v2 m v2
∵ F=ma , a= , ∴ F= /¿ (where F usually refers to T [tension])
r r
2
v 2
○ ***a= , F=mr ω
r
1
