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

In summary, the smallest value of h for which the sphere will not leave the track at the top of the loop is 2.7R - 1.7r, taking into account the force of norm exerted by the track on the sphere.
  • #1
ducnguyen2000
13
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


[tex]\Delta[/tex]PE = mgh - 2mgR
[tex]\Delta[/tex]KE = 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?
 
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  • #2
N is the force of norm exerted by the track on the cart at the top of the loop by the way
 
  • #3
haha, nvrmind, i figured out what i did wrong.
 

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

What is rotational energy in AP Physics C?

Rotational energy is the energy an object possesses due to its rotation. It is a form of kinetic energy and is dependent on the object's moment of inertia and angular velocity.

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The main difference between rotational energy and linear energy is the type of motion they are associated with. Rotational energy is associated with an object's rotation, while linear energy is associated with an object's linear motion. Additionally, rotational energy is dependent on an object's moment of inertia, while linear energy is dependent on an object's mass.

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The conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. In the case of rotational energy, if there are no external forces acting on an object, its total energy (including rotational energy) will remain constant. This means that as the object rotates, its potential energy may decrease, but its rotational energy will increase to compensate for it.

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