A small object slides along the frictionless loop-the-loop with a diameter of 3 m. What minimum speed must it have at the top of the loop in order to remain in contact with the loop?
A) 1.9 m/s
B) 3.8 m/s
C) 5.4 m/s
D) 15 m/s
E) 29 m/s

Question

A small object slides along the frictionless loop-the-loop with a diameter of 3 m. What minimum speed must it have at the top of the loop in order to remain in contact with the loop? 
A) 1.9 m/s 
B) 3.8 m/s 
C) 5.4 m/s 
D) 15 m/s 
E) 29 m/s

Quizplus · Accepted Answer

At the top of the loop, the normal force is pointing downwards while gravity is still pointing downwards. For the object to remain in contact with the loop, the normal force must equal zero i.e. gravity must be equal to the centripetal force:
$$mg = \frac{mv^2}{r}$$
where $m$ is the mass of the object, $v$ is the velocity at the top, $r$ is the radius of the loop. In this case, $r=\frac{3}{2}$ m.
Solving for $v$:
$$v = \sqrt{gr} = \sqrt{(9.8	ext{ m/s}^2)\left(\frac{3}{2}	ext{ m}ight)}\approx 3.8	ext{ m/s}$$
Therefore, the minimum velocity required at the top of the loop is 3.8 m/s.

A Small Object Slides Along the Frictionless Loop-The-Loop with a Diameter