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(II) A ball of radius $r_{0}$ rolls on the inside of a track of radius $R_{0}$ (see Fig. $61 ) .$ If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?

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$v=\sqrt{\left(\frac{10}{7} g\left(R_{0}-r_{0}\right)\right.}$

Physics 101 Mechanics

Chapter 10

Rotational Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

University of Michigan - Ann Arbor

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University of Sheffield

McMaster University

Lectures

02:21

In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

02:34

In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

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So we know the ball. The ball rose rolls without slipping. So the angular philosophies equaling the linear speed divided by are the radius of the ball. And so we can say that the energy at a equals the energy FT and simply used conservation of energy. We can say that the gravitational potential energy at a equals the potential energy at the plus, the final potential energy that be this would be equaling the potential energy at B plus the potential the kinetic energy of the center of mass at B plus the rotational, kinetic energy of B at point B. And so we can say m g are all right. You got to our equaling. And he are. So we're actually accounting for the gravitational potential energy off the ball even at point B. Um, and this will be plus 1/2 times mass sub d be mass times the velocity of B squared idea, translational kinetic energy, and then plus the rotational kinetic energy 1/2 times the moment of inertia at times the, uh, angular velocity at B squared. And so we can then say Hammond, she r equals on G are plus 1/2. Um these B squared plus 1/2 times the moment of inertia of a sphere 2/5 em are squared. And then this would be multiplied by the linear velocity at B, divided by the radius of the sphere squared. And so the velocity, the linear velocity at B is gonna be equaling 10 over seven times G multiplied by our minus R. This would be our linear speed at B again, the square root of 10 over seven times acceleration due to gravity times the radius of this curve minus the radius of the sphere itself. That is the end of the solution. Thank you for

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