Task
A mass point moves with a constant angular acceleration c1 on a circular path of radius R. How big must the angular acceleration be so that the mass point comes to a standstill after one revolution?
Given:
\[ \ddot \phi = c_1\]
\[\tag{IC 1} \dot \phi (t=0) = \omega_0\]
\[\tag{IC 2} \phi (t=0) = \phi_0\]
Solution
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Since it is a circular movement with a constant radius, the reduced consideration of the angle of rotation, the angular velocity and the angular acceleration is sufficient. The function is obtained by integrating the angular acceleration
\[\tag{1} \int \ddot \phi dt \rightarrow \dot \phi (t) = c_1 \cdot t + c_2\]
With the initial condition IC1 c2 can be solved.
\[\tag{2} \dot \phi (t = 0) = \omega_0\]
\[\tag{3} c_2 = \omega_0\]
\[\tag{4} \dot \phi (t) = c_1 \cdot t + \omega_0 \]
By integrating the angular velocity it follows
\[\tag{5} \int \dot \phi dt \rightarrow \phi (t) = \frac{1}{2} c_1 \cdot t^2 + \omega_0 t + c_3\]
Using the initial condition IC2 we can find c3:
\[\tag{6} \phi (t = 0) = \phi_0\]
\[\tag{7} c_3 = \phi_0\]
\[\tag{8} \phi (t) = \frac{1}{2} c_1 \cdot t^2 + \omega_0 t + \phi_0\]
The point in time at which the movement comes to a standstill is currently unknown and is referred to below as T. At time T, a complete revolution has taken place
\[\tag{9} \phi (t = T) = 2 \pi\]
\[\tag{10} 2 \pi = \frac{1}{2} c_1 T^2 + \omega_0 T + \phi_0\]
and the angular velocity is zero.
\[\tag{11} \dot \phi (t = T) = 0\]
\[\tag{12} 0 = c_1 T + \omega_0\]
\[\tag{13} c_1 = - \frac{\omega_0}{T}\]
The determined c1 will be used in equation 10 and solved for T.
\[\tag{14} T = \frac{2 \cdot (2 \pi - \phi_0)}{\omega_0} \]
T used in equation 13 yields the angular acceleration we are looking for.
\[\tag{15} c_1 = - \frac{\omega_0^2}{2 \cdot (2 \pi - \phi_0)} \]
