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At the moment `t=0` a point starts oscillating along the `x` axis according to the lasw` x=a sin omegat t`. Find: `(a)` the mean value of its velocity vector projection `( : v_(x) : )`, `(b)` the modulus of the mean velocity vector `|( : v : )|`, (c) the mean value of the velocity modulus `( : v : )` averaged over `3//8` of the period after the start. |
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Answer» As `x=a sin omega t` so,` v_(x)=a omega cos omega t` Thus, `lt v_(x) gt = int v_(x)dt//int dt=(int_(0)^((3)/(8)T)a omegacos(2pi//T)t dt)/((3)/(8)T)=(2sqrt(2)a omega)/(3pi)` (using `T=(2pi)/(omega))` `(b)` In accordance with the problem `vec(v)=v_(x)vec(i)`, so,`|lt vec(v) gt|=|lt v_(x) gt|` Hence, usning part (a) `|lt vec(v) gt|=|(2sqrt(2)a omega)/(3pi)|=(2 sqrt(2)a omega)/(3pi)` We have got, `v_(x)=a omega cos omegat` So, `{:(v=|v_(x)|=a omega cos omega t, "for", t leT//4),(= -aomega cos omega t,"for", T//4 le t3/8 T):}]` Hence, `lt v gt =(int f dt )/(int dt)(int _(0)^(T//4)a omega cos omegat dt+int_(T//4)^(3T//4)-a omega cos t dt )/(3T//8)` Using `omega=2pi//T`, an on evaluating the integral we get `lt v gt = (2(4-sqrt(2))a omega)/(3pi)` |
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