Velocity of the Center of Mass

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SUMMARY

The problem involves a spherical shell with a mass of 1 kg and a radius of 2.0 cm, released from a height of 1.00 m on an inclined plane. The velocity of the center of mass at the bottom can be determined using the conservation of energy principle, incorporating both translational and rotational kinetic energy equations: KE rotational (1/2 Iω²), potential energy (mgh), and KE translational (1/2 mv²). The solution requires manipulating these equations to express total energy in terms of mass, radius, and velocity, ultimately leading to a single formula with minimal unknowns.

PREREQUISITES
  • Understanding of conservation of energy principles
  • Familiarity with rotational dynamics and moment of inertia (I)
  • Knowledge of kinematics, specifically tangential acceleration (at = αR)
  • Basic algebraic manipulation skills for solving equations
NEXT STEPS
  • Study the conservation of energy in rolling motion
  • Learn about moment of inertia for different shapes, particularly spherical shells
  • Explore the relationship between angular velocity and linear velocity
  • Practice problems involving inclined planes and rolling objects
USEFUL FOR

Physics students, educators, and anyone interested in understanding the dynamics of rolling motion and energy conservation principles in mechanics.

ryankunzzz
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Homework Statement


[/B]
A shperical shell of mass = 1kg and radius 2.0cm is released from rest at the top of an inclined plane at a height of 1.00m. The ball rolls down the incline without slipping, what is the velocity of the center of mass at the bottom of the incline.

Homework Equations



I'm assuming it will be using

KE rotational, which is 1/2 IW^2
Potential Energy which is mgh
KE translational which is 1/2 mv^2
tangential acceleration: at= αR

The Attempt at a Solution



I have no idea where to begin. If we were given the angle of the incline, this would be much easier, but we are not.
 
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If you do not know the value of a parameter, leave it as a variable. HINT: use r for the radius of the ball.

Now see if you can manipulate the formula for rotational kinetic energy to eliminate "I" and "omega" leaving only m, r and v. The goal is to have a single formula for total energy with only a few unknowns in it (ideally only one).
 
I got it, thank you!
 

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