Applications of Circular Motion

Determining the tension on a ball swung in a vertical circle.  

If a ball is swung in a vertical circle, what would the tension be on the ball at the top, side and bottom of the path.  Rank these positions from least to greatest in terms of the tension force on the ball.  

Below how to solve for the tension in these three positions, first you need to draw free body diagrams for each position.  After drawing the free body diagrams, use Newton's Second Law to solve for tension in each case.  

Another example of circular motion is when a roller coaster, skateboarder or motorcycle rides through a vertical loop.   

I'd like to make a TV show where we would do calculations first to find out how fast you would need to go to safely make it through the top.  If you weren't going fast enough, you would just go to the side and try again.  Although, this wouldn't help the guy who made and and then crashed into the side.  

To solve for the required velocity, also known as the critical velocity.  You would need to determine where would be most dangerous.  This would be at the very top, if you can make it there, you can make it anywhere (just like New York City.) 

The work for this is shown below.  You would have to start with a free body diagram shown both Normal and Gravitational Force pointing down from the top.  Then using Newton's Second Law you have Normal and Gravity adding together to equal mass time acceleration since they both point in the direction of the acceleration (toward the center of the circle).  The tricky part is what to do with the Normal Force.  The idea is that the normal force is present if you are touching and if you are touching, you are not falling.  So, how much normal force do you need?  We'll all you need is *some* Normal Force, so as long as N>0, then your are touching and if you are touching, then you are not falling and if you are not falling you are happy.  

Lastly, we talked about your "weight".  Your weight is the value of the support force.  That could be the normal force is you are standing on the ground or the tension if you are hanging from a rope.  If you were in an elevator and someone cut the cord it would accelerate at 9.8m/s^2.  When you make a free body diagram for you in the elevator you would have Normal up and Gravity down.  Applying this in Newton's second law along with the acceleration of 9.8m/s^2, you will find the normal force equals zero.  This would give you a "weight" of zero.  Think if you were holding a bag and dropped it in a stationary elevator, if would fall to the ground.  But if you dropped it in the falling elevator, it would float, not fall to the floor because the floor is also falling.

Alternatively, If you were out in space and wanted to have an artificial gravity to give you weight, you could accelerate the rocket.  The free body diagram would have just normal force, no gravity.  But if you accelerated at 9.8m/s^2 then Newton's Second Law would yield the Normal Force to be mass times 9.8m/s^2, which is the same that we experience here on earth.