Answer:
The magnitude of the tangential velocity is ![v= 0.868 m/s](https://tex.z-dn.net/?f=v%3D%200.868%20m%2Fs)
The magnitude of the resultant acceleration at that point is ![a = 4.057 m/s^2](https://tex.z-dn.net/?f=a%20%3D%204.057%20m%2Fs%5E2)
Explanation:
From the question we are told that
The mass of the uniform disk is ![m_d = 40.0kg](https://tex.z-dn.net/?f=m_d%20%3D%2040.0kg)
The radius of the uniform disk is ![R_d = 0.200m](https://tex.z-dn.net/?f=R_d%20%3D%200.200m)
The force applied on the disk is ![F_d = 30.0N](https://tex.z-dn.net/?f=F_d%20%3D%2030.0N)
Generally the angular speed i mathematically represented as
![w = \sqrt{2 \alpha \theta}](https://tex.z-dn.net/?f=w%20%3D%20%5Csqrt%7B2%20%5Calpha%20%20%5Ctheta%7D)
Where
is the angular displacement given from the question as
![\theta = 0.2000 rev = 0.2000 rev * \frac{2 \pi \ rad }{1 rev}](https://tex.z-dn.net/?f=%5Ctheta%20%20%3D%200.2000%20rev%20%3D%200.2000%20rev%20%2A%20%5Cfrac%7B2%20%5Cpi%20%5C%20rad%20%7D%7B1%20rev%7D)
![=1.257\ rad](https://tex.z-dn.net/?f=%3D1.257%5C%20%20rad)
is the angular acceleration which is mathematically represented as
![\alpha = \frac{torque }{moment \ of \ inertia} = \frac{F_d * R_d}{I}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7Btorque%20%7D%7Bmoment%20%5C%20of%20%20%5C%20inertia%7D%20%20%3D%20%5Cfrac%7BF_d%20%2A%20R_d%7D%7BI%7D)
The moment of inertial is mathematically represented as
![I = \frac{1}{2} m_dR^2_d](https://tex.z-dn.net/?f=I%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20m_dR%5E2_d)
Substituting values
![I = 0.5 * 40 * 0.200^2](https://tex.z-dn.net/?f=I%20%3D%200.5%20%2A%2040%20%2A%200.200%5E2)
![= 0.8kg \cdot m^2](https://tex.z-dn.net/?f=%3D%200.8kg%20%5Ccdot%20m%5E2)
Considering the equation for angular acceleration
![\alpha = \frac{torque }{moment \ of \ inertia} = \frac{F_d * R_d}{I}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7Btorque%20%7D%7Bmoment%20%5C%20of%20%20%5C%20inertia%7D%20%20%3D%20%5Cfrac%7BF_d%20%2A%20R_d%7D%7BI%7D)
Substituting values
![\alph](https://tex.z-dn.net/?f=%5Calph)
![\alpha = \frac{(30.0)(0.200)}{0.8}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B%2830.0%29%280.200%29%7D%7B0.8%7D)
![= 7.5 rad/s^2](https://tex.z-dn.net/?f=%3D%207.5%20rad%2Fs%5E2)
Considering the equation for angular velocity
![w = \sqrt{2 \alpha \theta}](https://tex.z-dn.net/?f=w%20%3D%20%5Csqrt%7B2%20%5Calpha%20%20%5Ctheta%7D)
Substituting values
![w =\sqrt{2 * (7.5) * 1.257}](https://tex.z-dn.net/?f=w%20%3D%5Csqrt%7B2%20%2A%20%287.5%29%20%2A%201.257%7D)
![= 4.34 \ rad/s](https://tex.z-dn.net/?f=%3D%204.34%20%5C%20rad%2Fs)
The tangential velocity of a given point on the rim is mathematically represented as
![v = R_d w](https://tex.z-dn.net/?f=v%20%3D%20R_d%20w)
Substituting values
![= (0.200)(4.34)](https://tex.z-dn.net/?f=%3D%20%280.200%29%284.34%29)
![v= 0.868 m/s](https://tex.z-dn.net/?f=v%3D%200.868%20m%2Fs)
The radial acceleration at hat point is mathematically represented as
![= \frac{0.868^2}{0.200^2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B0.868%5E2%7D%7B0.200%5E2%7D)
![= 3.7699 \ m/s^2](https://tex.z-dn.net/?f=%3D%203.7699%20%5C%20m%2Fs%5E2)
The tangential acceleration at that point is mathematically represented as
![\alpha _t = R \alpha](https://tex.z-dn.net/?f=%5Calpha%20_t%20%3D%20R%20%5Calpha)
Substituting values
![\alpha _t = (0.200) (7.5)](https://tex.z-dn.net/?f=%5Calpha%20_t%20%3D%20%280.200%29%20%287.5%29)
![= 1.5 m/s^2](https://tex.z-dn.net/?f=%3D%201.5%20m%2Fs%5E2)
The magnitude of resultant acceleration at that point is
![a = \sqrt{\alpha_r ^2+ \alpha_t^2 }](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%7B%5Calpha_r%20%5E2%2B%20%5Calpha_t%5E2%20%7D)
Substituting values
![a = \sqrt{(3.7699)^2 + (1.5)^2}](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%7B%283.7699%29%5E2%20%2B%20%281.5%29%5E2%7D)
![a = 4.057 m/s^2](https://tex.z-dn.net/?f=a%20%3D%204.057%20m%2Fs%5E2)