Falling Masses, the Big Picture.

This lecture note will make your life ten-fold easier in the scope of the problems it addresses. Consider it a talisman. I discovered this a couple of weeks ago when I was solving these problems for my own conceptual understanding. So I waited till I can completely enunciate the big picture. When I confirmed that its valid for all the following problems I made this note and sharing with you.

I am not doing any derivation here. All of these are available in University physics — Young Freedman et al, Pearson Publication, which is my favorite text at its level, click on link to see it on Amazon.

University Physics, 13th Edition. Click on image to go to Amazon page of the book.

To make life simpler I am gleaning some relevant diagrams from across the web.

The falling mass (mass m) through a pulley (mass M, Moment of inertia: cMR^2) Photo-Credit: click to left.
The falling yo-yo (mass m/M, Moment of inertia: cMR^2) Photo-credit: click to left.
Rolling object (mass M, moment of inertia cMR^2) on incline with friction.  Photo-credit: click to left.

As such one needs to remember 9 different formula for these 3 different motions (of many different kinds of objects) 3 each, for velocity falling through height h, acceleration during this motion, tension (or in case of the incline, the friction).

I will tell some easy conceptual hacks which will help you remember everything (velocity, acceleration, tension/friction), for all the above problems, and you can even remember only two things: 1. the formula for velocity, acceleration, force for the simplest case of a mass falling under gravity 2. How to scale each of these to get the result for all the relevant problems. (mentioned above)

Here is the Big Picture:

  1. As shown in the first diagram, in case of the unwinding cable, there is a pulley of mass M, whose moment of inertia is cMR^2, with c taking the suitable value as per the shape of the pulley c = 1/2 (pulley is a disk). But it can take other values in other problems depending on the shape of the object, falling.  Falling mass in the problem of unwinding cable has mass m. Remember that the velocity of a mass m falling under gravity through height is simply: \sqrt{2gh}, its acceleration g and the force experienced is its mass times g = mg. All we need to do is scale the variable g suitably. g needs to be scaled down by factor 1+c\frac{M}{m}. For tension T which comes from the pulley whose mass is M and has a factor c in its moment of inertia its given by cM\times scaled acceleration.
  2. Yo-Yo is equivalent to the unwinding cable. We need to recognize that the falling mass takes the role of both the pulley and the falling mass in the problem of unwinding cable. Hence we need to set M = m and apply the rules of scaling to a simply falling mass, as we did in 1. Note that c is now from the moment of inertia of the falling object, hence the shape of the falling object determines its velocity, acceleration and tension in the cable.
  3. The Incline on which a sphere or cylinder would be rolling is also equivalent to the unwinding cable as well as the yo-yo, with the recognition M = m. Also there is an additional scaling of the acceleration due to angle of incline: sin\beta. There is no tension here but its role is played by the static friction acting at the instantaneous axis of rotation whence the rolling without slipping condition is satisfied.

Note: for the incline problem in the velocity expression there is no sin\beta factor as the incline’s total length has the same factor (h\to \frac{h}{sin\beta}) and it cancels the same in the scaling factor in g.

Take a pdf version (fallingmasses) of the following table, for better resolution and/or print.


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