A small bead of mass m=1g moves without friction along the inner surface of a deep spherical cup, which is motionless with respect to the Earth’s surface. The curvature of the cup has radius R=98 mm and center O. For ease of description, let’s pick a coordinate system with origin at the center of curvature of the cup, and a z axis pointing vertically down. a)The bead is deflected from equilibrium and released. Prove that the resulting motion is confined to a vertical plane (say the xz plane) containing the equilibrium and the initial position (point of release).(hint: use forces or torque & angular momentum) b)Prove explicitly that for small deviations from the equilibrium, the system in part “a” behaves as a simple harmonic oscillator (SHO) c)For the system in part “b”, derive the equation of motion x(t) based on the given initial conditions. Calculate the angular frequency, frequency, and period of these oscillations.
A small bead of mass m=1g moves without friction along the inner surface of a deep spherical cup, which is motionless with respect to the Earth’s surface. The curvature of the cup has radius R=98 mm and center O. For ease of description, let’s pick a coordinate system with origin at the center of curvature of the cup, and a z axis pointing vertically down.
a)The bead is deflected from equilibrium and released. Prove that the resulting motion is confined to a vertical plane (say the xz plane) containing the equilibrium and the initial position (point of release).(hint: use forces or torque &
b)Prove explicitly that for small deviations from the equilibrium, the system in part “a” behaves as a simple harmonic oscillator (SHO)
c)For the system in part “b”, derive the equation of motion x(t) based on the given initial conditions. Calculate the angular frequency, frequency, and period of these oscillations.
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