A particle moves in a circle of radius 4. 0 cm clockwise at constant speed of `2 cms^(-1)` , If ` hat(x) and hat(y)` are unit acceleration vectors along x-axis and y-axis respectively (in `cms^(-2)`) , find the acceleration of the particle at the instant half way between P and Q .

Explanation

As shown in the figure (ii) , let R be the midpoint of arc PQ . Then `angle PQR = 45^(@)`
Magnitude of acceleration at R ,
The acceleration a acts along RO
Magnitude of component of along x-axis
`a_(x) = a cos 45^(@)`
`= 1 xx (1)/(sqrt(2)) = (1)/(sqrt(2)) cms^(-2)`
`:. hat (a) _(x) = - (1)/(sqrt(2)) x^(2)`
Magnitude of component of a along y-axis,
`a_(y) = 1 xx sin 45^(@)`
`= (1)/(sqrt(2)) cms^(-2)`
`:. hat(a)_(y) = - (1)/(sqrt(2)) hat(y)`
Hence `hat(a) _(x) + hat(a)_(y) = (-1)/(sqrt(2)) (x^(2) + hat(y))`