Lecture25-2A

Lecture25-2A - Chapter 14 Static Equilibrium III Chapter 15 Oscillatory Motion II Example Slipping Sphere A uniform sphere is supported by a rope

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Chapter 14 Chapter 14 Static Equilibrium III Static Equilibrium III Chapter 15 Chapter 15 Oscillatory Motion II Oscillatory Motion II
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Example: Slipping Sphere Example: Slipping Sphere A uniform sphere is supported by a rope. The point where the rope is attached to the sphere is located so a continuation of the rope would intersect a horizontal line through the sphere’s center a distance R / 2 beyond the center, as shown. What is the smallest value for m s between the wall and the sphere? Taking torques about the contact between the sphere and the wall, assuming a mass m , yields: 3 2 RT cos30 mgR Summing forces in both the x and y directions: N T sin30 and s N T cos30 mg
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Example: Slipping Sphere Example: Slipping Sphere A uniform sphere is supported by a rope. The point where the rope is attached to the sphere is located so a continuation of the rope would intersect a horizontal line through the sphere’s center a distance R / 2 beyond the center, as shown. What is the smallest value for m s between the wall and the sphere? There are three unknowns, m s , T , and N . Substituting into the force equation in the y direction T from the torque equation and N from the x force equation: mg s sin30 cos30 2 mg 3cos30 3 2 s tan30 1 s cos30 2sin30 s cos30 3 /2 .866
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Example: Double Welled Potential Example: Double Welled Potential Consider the potential given by U x ax 2 2 bx 4 4 Find the equilibrium points and determine if they are stable. The equilibrium points are found from: dU dx ax bx 3 0 x 0, a / b Stability is determined by the second derivative of the potential at equilibrium. d 2 U dx 2 0 a 3 bx 2 | 0 a 0 unstable d 2 U dx 2 a / b a 3 bx 2 | a / b a 3 a 2 a 0 stable
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Simple Harmonic Motion Simple Harmonic Motion Mathematically such a force is described as: The is the force exerted by an ideal spring of spring constant k . From Newton’s 2 nd we can write: Simple harmonic motion results when an object is subject to a linear restoring force and is called simple harmonic motion , SHO. F kx F m d 2 x dt 2 kx An object experiencing such a force means that when it is displaced from equilibrium there is a force proportional to the distance from equilibrium that accelerates the object back towards its equilibrium position. How do we describe such motion?
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Simple Harmonic Motion Simple Harmonic Motion From Newton’s 2 nd : Simple harmonic motion results when an object is subject to a linear restoring force and is called simple harmonic motion , SHO. F
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This note was uploaded on 04/16/2008 for the course PHYS PHYS2A taught by Professor Hicks during the Spring '08 term at UCSD.

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Lecture25-2A - Chapter 14 Static Equilibrium III Chapter 15 Oscillatory Motion II Example Slipping Sphere A uniform sphere is supported by a rope

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