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 centripetal force required to keep the object moving in a circle is provided by its weight. Therefore, we can use the equation for centripetal force:

F = m * ac = m * v^2 / r

where m is the mass of the object, v is its speed, r is the radius of the loop (which is half its diameter), and ac is the centripetal acceleration.

At the top of the loop, the weight of the object is:

Fg = m * g

where g is the acceleration due to gravity. This force must be equal to the centripetal force:

Fg = m * g = m * v^2 / r

Solving for v:

v^2 = r * g
v = sqrt(r * g)

Substituting r = 1.5 m, and g = 9.81 m/s^2, we get:

v = sqrt(1.5 * 9.81) ≈ 3.8 m/s

Therefore, the minimum speed required at the top of the loop is 3.8 m/s, which corresponds to option B.

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