What is the Minimum Height for a Sphere to Stay on a Loop-the-Loop Track?

Click For Summary
SUMMARY

The minimum height \( h \) for a uniform solid sphere of radius \( r \) to stay on a loop-the-loop track of radius \( R \) is calculated to be \( h = 2.7R - 1.7r \). The initial attempt incorrectly concluded \( h = 2.7R \) due to miscalculations involving potential energy and kinetic energy equations. Key equations utilized include conservation of energy and the moment of inertia for a solid sphere. The correct analysis ensures that the normal force at the top of the loop is zero, confirming the sphere remains on the track.

PREREQUISITES
  • Understanding of conservation of energy principles
  • Familiarity with rotational dynamics and moment of inertia
  • Knowledge of kinematics, specifically circular motion
  • Ability to manipulate algebraic equations involving physical quantities
NEXT STEPS
  • Study the derivation of the conservation of energy in rolling motion
  • Learn about the moment of inertia for different shapes, focusing on solid spheres
  • Explore the dynamics of circular motion and the role of normal force
  • Investigate real-world applications of loop-the-loop physics in amusement park rides
USEFUL FOR

Physics students, educators, and engineers interested in dynamics and energy conservation principles, particularly in the context of rotational motion and circular tracks.

ducnguyen2000
Messages
13
Reaction score
0
1. A uniform solid sphere of radius r starts from rest at a height h and rolls without slipping along the loop-the-loop track of radius R as shown.
a) What is the smallest value of h for which the sphere will not leave the track at the top of the loop?


Attempt:

Homework Equations


\DeltaPE = mgh - 2mgR
\DeltaKE = 1/2 mv2 + 1/2 Iw2
ac = v2/R
v = rw

The Attempt at a Solution


mgh = 2mgR + 1/2 Iw2 + 1/2 mv2
I = 2/5 mr2
mgh = 2mgR + 1/5 mr2w2 + 1/2 mv2
N = mg - mac = mg - mv2/R = 0
v2 = gR
mgh = 2mgR + 1/2 mv2 + 1/5 mv2 +2mgR = 2.7mgR
h = 2.7R

however, this answer is wrong, and the correct one is h = 2.7R - 1.7r. can anybody correct me on this?
 
Physics news on Phys.org
N is the force of norm exerted by the track on the cart at the top of the loop by the way
 
haha, nvrmind, i figured out what i did wrong.
 

Similar threads

Loop the Loop Problem Solution | N > 0, Energy Conservation Method
  • · Replies 8 ·
Replies
8
Views
4K
Starting height of marble rolling around a loop the loop
  • · Replies 12 ·
Replies
12
Views
12K
Conservation of energy problem: Ball rolling down inclined plane and then through a loop-the-loop
  • · Replies 10 ·
Replies
10
Views
3K
What Is the Minimum Release Height for a Rolling Ball on a Loop-the-Loop Track?
  • · Replies 1 ·
Replies
1
Views
4K
Solving a Physics Problem: Find Velocity of Block at Bottom of Loop
  • · Replies 1 ·
Replies
1
Views
2K
How High Will the Puck Shoot Up from the Plate?
  • · Replies 27 ·
Replies
27
Views
4K
Minimum h for Car to Remain on Track Around Loop
  • · Replies 6 ·
Replies
6
Views
2K
Kinetic / Potential Energy Problem
  • · Replies 4 ·
Replies
4
Views
8K
What Minimum Height Ensures a Marble Stays on a Loop-the-Loop Track?
  • · Replies 14 ·
Replies
14
Views
24K
What Is the Minimum Height for a Ball to Complete a Loop in Physics?
  • · Replies 24 ·
Replies
24
Views
7K