Physics

Asked by Dnyanutheurkar | 12th Sep, 2021, 02:59: PM

Expert Answer:

Radius vector begin mathsize 14px style r with rightwards arrow on top space equals space left parenthesis space 10 space cos omega t space i with hat on top space minus space 10 space sin omega t space j with hat on top space right parenthesis space m end style  ............................ (1)
Magnitude of radius vector begin mathsize 14px style open vertical bar r with rightwards arrow on top close vertical bar space equals space square root of 100 left parenthesis cos squared omega t space plus space sin squared omega t right parenthesis end root space equals space 10 space m end style ......................(2)
velocity begin mathsize 14px style v with rightwards arrow on top space equals space fraction numerator d r with rightwards arrow on top over denominator d t end fraction equals omega left parenthesis space minus 10 space sin omega t space i with hat on top space space minus space 10 space cos omega t space j with hat on top space right parenthesis space m divided by s end style .......................(3)
magnitude of velocity begin mathsize 14px style open vertical bar v close vertical bar space equals space omega square root of 100 left parenthesis sin squared omega t space plus space cos squared omega t space right parenthesis end root space equals space 10 space omega space m divided by s end style .........................(4)

Acceleration begin mathsize 14px style a with rightwards arrow on top space equals space fraction numerator d space v with rightwards arrow on top over denominator d space t end fraction space equals space omega squared space left parenthesis negative 10 space cos omega t space i with hat on top space plus space 10 space sin omega t space j with hat on top space right parenthesis space equals space minus omega squared space r with rightwards arrow on top space m divided by s squared end style ........................(5)
Magnitude of acceleration begin mathsize 14px style open vertical bar a with rightwards arrow on top close vertical bar space equals space omega squared open vertical bar r with rightwards arrow on top close vertical bar space equals space 10 omega squared space m divided by s squared end style ....................... (6)
 
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Statement-1 :- Centripetal acceleration at t = 1 s is 10ω2
 
Above statement is true as per eqn.(5) and (6) .
Acceleration is directed in direction of radius vector and its magnitude is 10ω2 i.e. independent of time
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Statement-2 :- Centripetal acceleration at t = 2 s is 40ω2
 
Above statement is false because magnitude of acceleration is always 10ω2 and is independent of time.
 
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Statement-3 :- Speed of particle is constant
 
Above statement is true because speed is magnitude of velocity that is constant 10ω as per eqn.(4)
 
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Statement-4 :- Acceleration is directed towards origin
 
Above statement is true. we determined acceleration  in eqn.(5) .
we see that direction of acceleration is in the direction of radius vector.
 
Negative sign shows acceleration is directed towards origin.

Answered by Thiyagarajan K | 12th Sep, 2021, 04:14: PM