CIRCULAR MOTION & WORK POWER ENERGY A body moving with constant speed in a circular path is continuously accelerated towards the centre of rotation. The magnitude of this normal acceleration is given by v2 = ω2 r an = r where v is the constant speed (v = ωr) and r is the radius of the circular path
Tangential area : at =
dv , dt
at 2 + a 2n
a= v2 an
2.
Radius of curvature : r =
3.
According to Newton’s second law, a body moving in a circular path with constant speed must be acted upon by an unbalanced force which is always directed towards the centre. This necessary unbalanced force is called the centripetal force. mv 2 F= = mω2r r
4.
Centrifugal force is a pseudo force which is observed an observer in rotating frame. 2 Fcf = mωframe r Work (W) : The work W done by a constant force F when its point of application undergoes a displacement s is defined as W = F.s = Fs cos θ where θ is the angle between F and s.Work is a scalar quantity and its SI units is N-m or joule (J).
Note: Only the component (F cos θ) of the force F which is along the displacement contributes to the work done. If F = F ˆi + F ˆj + F kˆ and s = ∆xˆi + ∆yˆj + ∆zkˆ x
then 5.
y
z
W = F ·s = Fx∆x + Fy∆y + Fz ∆ z
Work done by a Variable Force : When the magnitude and direction of a force varies with position, The work done by such a force for an infinitesimal displacement ds is given by dW = F · d s
In terms of rectangular components,
WAB =
6.
XB
YB
ZB
XA
YA
ZA
∫ Fx dx + ∫ Fydy + ∫ Fzdz
Work Done by a Spring Force : The work done by the spring force for a displacement from xi to xf is given by 1 Ws = − k x f2 − x i2 2
(
)
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Page 9 of 28 PARTICLE DYNAMICS
1.