Period of Oscillations near equator

In summary, at latitude Φ, the bead experiences a centrifugal force and a net force in the direction of the wire.
  • #1
Decadohedron
18
0

Homework Statement


A bead slides along a frictionless wire which lies in the N/S direction, midpoint at the equator. All points along the wire are the same distance from the center of the earth. The bead is initially at rest then released a small distance, δ, to the north of the equator. Because of effective g doesn't point directly towards the Earth's center, there is a small component along the wire that always points back towards the equator. This means that when released, the bead will oscillate back and forth just like a mass on a spring. What is the period of these oscillations?

Homework Equations


Coriolis force would be 0 as the bead is fixed on a wire, and the centrifugal force would be the force always pointing it back towards the equator - thus the formula being

1.
-GMm/r2 + mω2r = 0

2.
T = 2π√(m/k); ω2=m/k

The Attempt at a Solution



I was going to solve for ω and then just plug into the Period formula of the mass-spring system. But it seems overly simple and feels like I'm missing something.

Am I on the right track or should I be thinking about this differently??

Thanks.
 
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  • #2
Decadohedron said:
-GMm/r2 + mω2r = 0
It isn't in orbit.
 
  • #3
haruspex said:
It isn't in orbit.

So I should be using mr'' = -GMm/r2 + mω2r instead and solve the PDE and go from there?
 
  • #4
Decadohedron said:
So I should be using mr'' = -GMm/r2 + mω2r instead and solve the PDE and go from there?
What about the forces from the wire?
 
  • #5
haruspex said:
What about the forces from the wire?

There would be the parallel component in the r direction mω2rcos2(Φ) + the perpendicular component mω2rsin(Φ)cos(Φ)
 
  • #6
Decadohedron said:
There would be the parallel component in the r direction mω2rcos2(Φ) + the perpendicular component mω2rsin(Φ)cos(Φ)
You don't need to calculate the forces from the wire, just take them into account in your analysis.

Draw a diagram in the NS, up-down plane showing the bead at latitude Φ. What forces act (centrifugal being a fictitious force)? What is the net force?
 

Related to Period of Oscillations near equator

1. What is the Period of Oscillations near the equator?

The Period of Oscillations near the equator refers to the time it takes for a system or object to complete one full oscillation or cycle of movement near the Earth's equator. In other words, it is the time between when an object starts at a certain point, moves away, and then returns to its starting point near the equator.

2. What causes the Period of Oscillations near the equator?

The Period of Oscillations near the equator is largely caused by the Earth's rotation and the Coriolis effect. The Earth's rotation causes objects near the equator to experience a centripetal force, which results in a circular motion. The Coriolis effect, on the other hand, causes objects to deflect to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, creating an oscillating motion near the equator.

3. How is the Period of Oscillations near the equator measured?

The Period of Oscillations near the equator can be measured in seconds, minutes, or hours, depending on the specific system or object being observed. It can be measured using instruments such as a stopwatch or a timer, or through mathematical calculations based on the known parameters of the system.

4. What are some examples of systems that exhibit Period of Oscillations near the equator?

Some examples of systems that exhibit Period of Oscillations near the equator include pendulums, Foucault pendulums, and the Coriolis effect on winds and ocean currents. These systems all experience oscillatory motion due to the Earth's rotation and the Coriolis effect near the equator.

5. Why is the Period of Oscillations near the equator important to study?

The Period of Oscillations near the equator is important to study because it has significant effects on various natural phenomena, such as weather patterns, ocean currents, and the Earth's climate. Understanding the Period of Oscillations near the equator can also help in predicting and mitigating potential natural disasters, such as hurricanes and typhoons, which are influenced by oscillatory motion near the equator.

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