The pendulum shown in figure is pulled to a 10° angle on the left side and released.
a. What is the period of this pendulum?
b. What is the pendulum’s maximum angle on the right side?
a. The time period of the pendulum is
b. The pendulum's maximum angle on the right side is
The pendulum is pulled to a angle on the left side and released. So, when we calculating the time period we have to sum up the two time periods of the two ends of pendulum of length .
The time period of the pendulum written as
L is the length of the pendulum and
g is the acceleration due to gravity.
The total time period is equal to the sum of periods of two ends of the pendulum
The pendulum pulled to a angle on the left side. We have to find the pendulum maximum angle at the right side.
For that we have to use the formula of conservation of energy of simple pendulum. Where, as a pendulum swings, its potential energy changes to kinetic energy, then back to potential energy, then back to kinetic energy.
Therefore, the equation of conservation of energy
where is the length of the pendulum
Suppose a large spherical object, such as a planet, with radius and mass has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius ; there is no net gravitational force from the mass in the spherical shell with .
. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of , , , , and any necessary constants.
. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with . Suppose an intrepid astronaut exploring a -km-diameter, kg asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?
A penny rides on top of a piston as it undergoes vertical simple
harmonic motion with an amplitude of 4.0cm . If the frequency
is low, the penny rides up and down without difficulty. If the
frequency is steadily increased, there comes a point at which the
penny leaves the surface
a. At what point in the cycle does the penny first lose contact
with the piston?
b. What is the maximum frequency for which the penny just
barely remains in place for the full cycle?
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