a point traversed half a circle of radius R=160cm during time interval t=10s.calculate the following quantities averaged over that time
(a)the mean velocity <v>
(b)the modulus of the mean velocity vector |<v>|
(c)the modulus of the mean vector of the total acceleration |<w>| if the point moved with constant tangent acceleration.

Asked by JAIBHARGAVA200 | 4th Nov, 2014, 04:16: PM

Expert Answer:

begin mathsize 14px style Given : straight R equals 160 space cm time space interval equals 10 space straight s left parenthesis straight a right parenthesis space mean space velocity space open angle brackets straight v close angle brackets equals fraction numerator Displacement over denominator time space interval end fraction equals fraction numerator 2 straight R over denominator straight t end fraction equals fraction numerator 2 cross times 160 over denominator straight t end fraction equals 32 space cm divided by straight s left parenthesis straight b right parenthesis space Modulus space of space the space mean space velocity space open vertical bar open angle brackets straight v with rightwards arrow on top close angle brackets close vertical bar equals fraction numerator Total space path space travelled over denominator time space interval end fraction equals πR over straight t equals fraction numerator 3.14 cross times 160 over denominator straight t end fraction equals 50.24 space cm divided by straight s left parenthesis straight c right parenthesis space The space modulus space of space mean space vector space of space total space acceleration space open vertical bar open angle brackets straight omega close angle brackets close vertical bar space space space space space space We space have space eq comma straight omega squared equals straight omega subscript 0 superscript 2 plus 2 αθ minus minus minus space minus space left parenthesis straight i right parenthesis where comma space straight omega subscript 0 equals initial space angular space velocity space space space space space space space space space space space straight omega equals final space angular space velocity space space space space space space space space space space space space straight alpha equals angular space acceleration space space space space space space space space space space space space straight theta equals angular space displacement since space straight w subscript 0 equals 0 space & space point space travelled space half space circle space rightwards double arrow straight theta equals straight pi hence space eq comma space left parenthesis straight i right parenthesis space becomes straight omega squared equals 2 αθ equals 2 απ We space have space eq. space of space motion straight theta equals straight omega subscript 0 straight t plus 1 half αt squared equals 1 half αt squared minus minus minus minus left parenthesis straight omega subscript 0 equals 0 right parenthesis straight pi equals 1 half αt squared rightwards double arrow straight alpha equals fraction numerator 2 straight pi over denominator straight t squared end fraction Relation space between space linear space velocity space and space angular space velocity space is space given space as straight v equals Rω equals straight R fraction numerator 2 straight pi over denominator straight t end fraction The space modulus space of space mean space vector space of space total space acceleration space open vertical bar open angle brackets straight omega with rightwards arrow on top close angle brackets close vertical bar equals fraction numerator straight v subscript straight f minus straight v subscript straight i over denominator straight t end fraction equals fraction numerator begin display style fraction numerator 2 πR over denominator straight t end fraction end style minus 0 over denominator straight t end fraction equals fraction numerator 2 πR over denominator straight t squared end fraction space open vertical bar open angle brackets straight omega with rightwards arrow on top close angle brackets close vertical bar equals fraction numerator 2 cross times 3.14 cross times 160 over denominator 10 squared end fraction equals 10.05 space cm divided by straight s squared end style

Answered by Priyanka Kumbhar | 5th Nov, 2014, 11:42: AM

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