Uniform Circular Motion

Question

Show that the magnitude of the velocity for an object in uniform circular motion with:

x(t)=Rcos(V_0*t/R) and y(t)=Rsin(V_0*t/R) as position equations equals V_0.

 

And determine the components of the acceleration of an object in uniform circular
motion with the same equations. Show that the acceleration is equal to v₀²/R.

Answer 1

v = √(v_x² + v_y²)

 

v_x = dx/dt = -v₀ sin(v₀t/R)

 

v_y = dy/dt = v₀ cos(v₀t/R)

 

v² = v₀² [sin²(v₀t/R) + cos²(v₀t/R)] = v₀²(1) = v₀²

 

v = v₀

 

a_x = dv_x/dt = -v₀²/R cos(v₀t/R)

 

a_y = dv_y/dt = -v₀²/R sin(v₀t/R)

 

a² = a_x² + a_y² = (v₀²/R)² [cos²(v₀t/R) + sin²(v₀t/R)]

 

a² = (v₀²/R)² (1) = (v₀²/R)²

 

a = v₀²/R