What happens to circular motion when the radius changes?

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In circular motion, increasing the radius results in smaller centripetal acceleration, while decreasing the radius leads to larger centripetal acceleration. Centripetal force is essential for maintaining circular motion, and the centrifugal force acts as its reaction, enabling objects like toy cars to complete loops. The vector sum of gravitational and centrifugal accelerations must direct towards the center for successful motion. The formula for centripetal acceleration is Ac = v^2/r, where v is velocity and r is the radius. Understanding these dynamics is crucial for analyzing energy in uniform circular motion and the mechanics of spinning objects.
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what will happen in circular motion if the radius increase, and if it decrease.

what effects and causes that will make an abject successfuly complete its circular motion in increasing radius (like car toy loop)

for the first question my answer is if the radius increase the centrpetal acceleration will get smaller and if the radius decrease the centrpetal acceleration will get bigger. but i don't know what roll does the centrpetal acceleration have on circular motion.
 
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It's the centrifugal force (the reaction to the centripetal force) that enables the toy car to complete the loop. The centrifugal acceleration also increases when the radius of the circular motion decreases. So if the vectorial sum of the gravitational and the centrifugal acceleration doesn't point away from the "road" then the toy car will complete the loop.
 
What are the equations for centripital acceleration (in terms of omega or velocity)? What is the energy in uniform circular motion? What does an ice skater do in a spin to speed up the spin?
 
the formula that i know is centripetal acceleration (Ac)=v^2/r
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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