Please show all work and explain answer. A certain spinning wheel is known to have a non-constant angular acceleration given bya(0) = -k0%3Dso that it depends on the angular location of the wheel. When it spins in the positive direction itbegins to angularly accelerate negatively back to zero, and when it spins in the negative directionit begins to positively accelerate back to zero.An example could be a wheel with a charged insulated ball attached to the edge in a constantelectric field.positive net chargeUniform Electric Field GeneratorWrite a function for the angular location of this wheel as a function of time given that at timet=0, you start the wheel at a small angle away from zero equal to 00, and with no initial angularvelocity.

This problem is an example of angular simple harmonic motion where the force experienced by the particle is always directed towards the origin. If the angular acceleration is known then the angular displacement/position is given by α=d2θdt2 Therefore if the angular acceleration is known in terms of θ and is directly proportional to θ such that α=mθ Then the differential equation of motion is given by d2θdt2=mθ This a second-order ordinary differential and can be solved by the trial solution method. In the given question we have the angular acceleration a function of angular displacement αθ=-kθ where k is a constant. Therefore the differential equation is given by d2θdt2=-kθ⇒d2θdt2+kθ=0 To solve this differential equation let us consider the trial solution θ=emt. Thus dθdt=memt⇒d2θdt2=m2emt Substituting these values in the equation we get emtm2+k=0⇒m=±ki The roots are complex. Therefore the solution of the differential equation is θ=Acoskt+Bsinkt Now A and B are undetermined and can be found out using the boundary conditions. The boundary conditions available to us are ωt=0=0⇒dθt=0dt=0 and θt=0=θ0 By the first boundary condition we have dθdt=-Aksinkt+Bkcoskt⇒dθt=0dt=B=0 Thus we have θ=Acoskt Using the second boundary condition we have A=θ0 Thus the complete solution is given by θ=θ0coskt Answer: The angular position function is θ=θ0coskt

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Advanced Physics

Write a function for the angular location of this wheel as a function of time given that at time t=0, you start the wheel at a small angle away from zero equal to 00, and with no initial angular velocity.

Write a function for the angular location of this wheel as a function of time given that at time t=0, you start the wheel at a small angle away from zero equal to 00, and with no initial angular velocity.