03032009

03032009 - Kinematics Motion of a Particle A reference...

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Kinematics – Motion of a Particle A reference frame is a coordinate system specifying the positions of points by specifying the components of the position vector r relative to the origin. Suppose P is in a motion relative to the chosen reference frame, so the r is a function of time t r = r(t ) describes the motion of P

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Velocity of Particle in Motion The velocity of P relative to the given reference frame at time t is defined by ) 1 . 13 ( ) ( r ) ( r lim r v 0 t t t t dt d t Δ Δ + = = Δ where the vector r(t+ Δ t) – r(t) is the change in position, or displacement, of P during the interval of time Δ t Therefore, the velocity v is the rate of change of position of P.
Acceleration of Particle in Motion The acceleration of P relative to the given reference frame at time t is defined by ) 2 . 13 ( ) ( v ) ( v lim v a 0 t t t t dt d t Δ Δ + = = Δ where v(t+ Δ t) – v(t) is the change in velocity of P during the interval of time Δ t . Acceleration is the rate of change of the velocity of P at time t i.e. ) / ( r r 2 2 2 s m dt d or dt d dt d

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Reference Frame for Motion of Particle Let O’ be an arbitrary fixed point and r’ be the position vector from O’ to P dt d ' r ' v = Velocity of P relative to O’ is Velocity of P relative to O is dt d r v = dt d ' r ' v =
Reference Frame for Motion of Particle v r R r ' r ' v = = = = dt d dt d dt d dt d Let R be the vector from O to O’ such that r = R + r’ or r’ = r – R R is a constant, therefore, velocity of P relative to O’ is Similarly, for acceleration dt d and dt d v a ' v ' a = = Therefore, v’ = v and a’ = a Î The velocity and acceleration of a point P relative to a given frame of reference do not depend on the location of the fixed reference point used to specify the position of P.

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