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depict two circulat mothions. The radius of the circle, the period of revolution, the initial position and sense of revolution are indicated in the figures. Obtain the simple harmonic motinosof the x-projection of the radius vector of the rotating particle P in each case. |
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Answer» Refere to figure , `T=4s, ` intial pahse, `phi=45^(@)=(pi)/(4)rad` with the positive direction of x-axis. After a time t, it covers an angle `(2pi)/(T)t` in the anitclockwise direction and makes an angle `((2pi)/(T)t+(pi)/(4))` with the x-axis. The projectino of OP on the x-axis at time t is `x=acos((2ppit)/(T)+phi)=acos((2pit)/(4)+(pi)/(4))` Refer to figure `T=30s` inital phase, `phi=90^(@)=(pi)/(2)rad` with the x-axis. After a time t, it covers an angle `(2pit)/(T)` in the clockwise direction and makes an angle `((pi)/(2)-(2pit)/(T))` with the x-axis. Now, the projection of OP on the x-axis at time t is given by `x=bcos((pi)/(2)-(2pit)/(T))=bcos((2pit)/(T)-(pi)/(2))` `b cos((2pit)/(30)-(pi)/(2))=bcos((pit)/(15)-(pi)/(2))` |
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