Answer:
(a). The magnitude of the total acceleration of the ball is 239.97 m/s².
(b). The angle of the total acceleration relative to the radial direction is 11.0°
Explanation:
Given that,
Radius of the circle = 0.681 m
Angular acceleration = 67.7 rad/s²
Angular speed =18.6 rad/s
We need to calculate the centripetal acceleration of the ball
Using formula of centripetal acceleration
![a_{c}=\omega^2\times r](https://tex.z-dn.net/?f=a_%7Bc%7D%3D%5Comega%5E2%5Ctimes%20r)
Put the value into the formula
![a_{c}=(18.6)^2\times0.681](https://tex.z-dn.net/?f=a_%7Bc%7D%3D%2818.6%29%5E2%5Ctimes0.681)
![a_{c}=235.5\ m/s^2](https://tex.z-dn.net/?f=a_%7Bc%7D%3D235.5%5C%20m%2Fs%5E2)
We need to calculate the tangential acceleration of the ball
Using formula of tangential acceleration
![a_{t}=r\alpha](https://tex.z-dn.net/?f=a_%7Bt%7D%3Dr%5Calpha)
Put the value into the formula
![a_{t}=0.681\times67.7](https://tex.z-dn.net/?f=a_%7Bt%7D%3D0.681%5Ctimes67.7)
![a_{t}=46.104\ m/s^2](https://tex.z-dn.net/?f=a_%7Bt%7D%3D46.104%5C%20m%2Fs%5E2)
(a). We need to calculate the magnitude of the total acceleration of the ball
Using formula of total acceleration
![a=\sqrt{a_{c}^2+a_{t}^2}](https://tex.z-dn.net/?f=a%3D%5Csqrt%7Ba_%7Bc%7D%5E2%2Ba_%7Bt%7D%5E2%7D)
Put the value into the formula
![a=\sqrt{(235.5)^2+(46.104)^2}](https://tex.z-dn.net/?f=a%3D%5Csqrt%7B%28235.5%29%5E2%2B%2846.104%29%5E2%7D)
![a=239.97\ m/s^2](https://tex.z-dn.net/?f=a%3D239.97%5C%20m%2Fs%5E2)
(b). We need to calculate the angle of the total acceleration relative to the radial direction
Using formula of the direction
![\theat=\tan^{-1}(\dfrac{a_{t}}{a_{c}})](https://tex.z-dn.net/?f=%5Ctheat%3D%5Ctan%5E%7B-1%7D%28%5Cdfrac%7Ba_%7Bt%7D%7D%7Ba_%7Bc%7D%7D%29)
Put the value into the formula
![\theta=\tan^{-1}(\dfrac{46.104}{235.5})](https://tex.z-dn.net/?f=%5Ctheta%3D%5Ctan%5E%7B-1%7D%28%5Cdfrac%7B46.104%7D%7B235.5%7D%29)
![\theta=11.0^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3D11.0%5E%7B%5Ccirc%7D)
Hence, (a). The magnitude of the total acceleration of the ball is 239.97 m/s².
(b). The angle of the total acceleration relative to the radial direction is 11.0°