A small solid marble of mass m and radius r rolls without slipping along the tra

Question

A small solid marble of mass m and radius r rolls without slipping along the track (similar

to the "loop-the-loop" circus constructions) shown in the figure. The marble was released

from rest somewhere on the straight section of the track. What is the minimum height h

at which the marble can be released so that it does not leave the track at the top of the

loop? The radius of the loop R is much larger than r. The moment of inertia of a sphere is

2mr2=5.

A small solid marble of mass m and radius r rolls without slipping along the track (similar to the "loop-the-loop" circus constructions) shown in the figure. The marble was released from rest somewhere on the straight section of the track. What is the minimum height h at which the marble can be released so that it does not leave the track at the top of the loop? The radius of the loop R is much larger than r. The moment of inertia of a sphere is 2mr2=5.

Explanation / Answer

at top of loop

mg + N = mv^2/R

min speed N = 0

v = sqrt(gR)

conservation of energy

mgh = 1/2 mv^2 + 1/2 I w^2 + m g 2R

m g h = 1/2 m v^2 + 1/2 *2/5 m r^2 w^2+ m g 2 R

plug in for v and use that wr = v

m g h = 1/2 m g R + 1/5 m g R + m g 2 R

h = 1/2 R + 1/5 R + 2R = 5/10 R + 2/10 R + 20/10 R = 27/10 R

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.