- Since both mg and Ftid are conservative (both gravity forces), we can write them as the gradient of a potential energy:
- where Ueg is the potential due to Earth gravity and Utid is that due to tidal forces.
- By inspection of , the latter is
- Referring to the figure at right, since U = Ueg + Utid has to be constant on the surface, or
- But Here h is the height difference between P and Q.
- To determine the quantity we just evaluate Utid at P and Q.
Magnitude of the Tides-2 - Putting all of this together, we have
- Finally, since we have
- Putting in numbers (Mmoon = 7.35x1022 kg, Me = 5.98x1024 kg, Re = 6.37x106 m, and do = 3.84x108 m), we find the height of the tides due to the Moon alone:
- You can easily use this same expression, substituting values for the Sun, to find
- When the Sun and Moon align, you get spring tides (this term has nothing to do with the season), and the two add to give h = 54 + 25 = 79 cm. When the Sun and Moon are at right angles, you get neap tides and the two subtract to give h = 54 – 25 = 29 cm.
- These same tidal forces act elsewhere in the solar system, e.g. Io.
9.3 Angular Velocity Vector - We are now going to discuss accelerating frames where the non-inertial frame is rotating (relative to the inertial frames).
- Before we can discuss these, we must introduce some concepts and notation for handling rotation.
- Many rotation problems involve axes fixed in a rigid body (e.g. the rotation of the Earth about its spin axis). We will see several other examples in Chapter 10. We can cast the problem into one in which the rotation is about a fixed axis by considering either a real fixed axis (i.e. a pendulum rotating about its fixed pivot, or a wheel about its axle) or for other bodies like a spinning baseball we simply switch to its CM frame.
- Euler’s theorem, which we will state without proof, is that the most general motion of any body relative to a fixed point O is a rotation about some axis through O. (See Goldstein, Poole & Safko for a proof.)
- To specify this rotation about a given point O, we only have to give the direction of the axis and the rate of rotation, or angular velocity . Because this has a magnitude and direction, it is an obvious choice to write this rotation vector as , the angular velocity vector. It points in the right-hand rule sense.
Do'stlaringiz bilan baham: |
