Vertical loop
How high must a roller coaster start to make it over the top of the loop?
At the top of the loop, gravity and the normal force from the track together provide the centripetal force. The normal force cannot be negative, so the speed at the top must be at least √(gR). Conservation of mechanical energy, mgh = mg·2R + ½mv², then shows that the ball must be released from rest at a height of at least 2.5R. If the height is too low, the ball leaves the track where the normal force drops to zero and continues as a projectile.
Normal force at top vs. release height
The normal force is zero at the critical height 2.5R; a negative value means the ball has already left the track before reaching the top.
Think about it
With a loop radius of 1 m, can a ball released from a height of 2.4 m make it over the top?
No. The critical height is 2.5R = 2.5 m, and 2.4 m is below it, so the ball leaves the track before reaching the top. Set the height to 2.4 m and watch where it leaves the track.
Model assumptionsFrictionless track · Ball treated as a point mass · Released from rest
Data log
Set the parameters and click “Record data”. Record several rows with different parameters to plot their relationship and fit a straight line; pause and record at different times to study how a reading changes with time.