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A body is moving under the acton of cent...

A body is moving under the acton of central force `vrc(F)(r ) hat(e )`. Then, choose the correct statement (symbols are having usual meaning and `hat(e ), hat(e )`, denote unit vectors along the radial and tangential direction, respectively ) from the following.

A

`vec(v) = (dr)/(dt) hat(e)_(r ) + r(d theta)/(dt) hat(e)_(theta), vec(a) = [(d^(2)r)/(dt^(2)) - r ((d theta)/(dt))^(2)] hat(e)_(r )+ [2(dr)/(dt) (d theta)/(dt) + r (d^(2)theta)/(dt^(2))]hat(e)_theta `

B

`vec(v) = (dr)/(dt) hat(e)_(r ) + r(d theta)/(dt) hat(e)_(theta), vec(a) = [(d^(2)r)/(dt^(2)) + r ((d theta)/(dt))^(2)] hat(e)_(r )+ [2(dr)/(dt) (d theta)/(dt) - r (d^(2)theta)/(dt^(2))]hat(e)_theta `

C

`vec(v) = (dr)/(dt) hat(e)_(r ) - r(d theta)/(dt) hat(e)_(theta), vec(a) = [(d^(2)r)/(dt^(2)) + r ((d theta)/(dt))^(2)] hat(e)_(r )+ [2(dr)/(dt) (d theta)/(dt) + r (d^(2)theta)/(dt^(2))]hat(e)_theta`

D

`vec(v) = (dr)/(dt) hat(e)_(r ) - r(d theta)/(dt) hat(e)_(theta), vec(a) = [(d^(2)r)/(dt^(2)) - r ((d theta)/(dt))^(2)] hat(e)_(r )+ [2(dr)/(dt) (d theta)/(dt) + r (d^(2)theta)/(dt^(2))]hat(e)_theta`

Text Solution

Verified by Experts

The correct Answer is:
A
Use `(dhat(e)_(r ))/(dt) = (d theta)/(dt) hat(e)_(theta)`. and `(dhat(e)_(theta))/(dt) = -(dtheta)/(dt) hat(e)_(r )`
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