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Symmetry.

Problem solving

Four methods to check and predict result

1. Dimensional analysis

2. Extreme cases

3. Symmetry

4. Credible values

Let us the height of projectile as an example:

(1)

Is it true?

Dimensional analysis shows that:

There is mistake in the result (1).

May be ?

(2)

Extreme case: It is evident that if ,

But from (2) we have !

There is mistake in the result (2).

May be ? (3)

Let us check the following extreme case:

We have the projectile thrown vertically.

From the energy conservation law one can conclude:

and

And the factor 2 was missing in the denominator (3).

Symmetry.

Problem: the bus makes one half pass S with the speed and another half pass with the speed . What is the average speed of this travel?

Let

Are there any mistakes at the result?

String Oscillations (Moscow Phys-Tech)

The frequency of oscillation of a string depends on its length L, the force applied to its ends T, and the linear mass density . Using dimensional analysis, find this dependence.

Falling Chain (MIT, Stanford)

A chain of mass M and length L is suspended vertically with its lower end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length x of the chain has fallen? Neglect the size of the individual links.

Cat and Mouse Tug of War (Moscow Phys-Tech, MIT)

A rope is wrapped around a fixed cylinder as shown in Figure P. 1.2. There is friction between the rope and the cylinder, with a coefficient of friction ;

The angle defines the arc of the cylinder covered by the Pulling Strings (MIT) rope. The rope is much thinner than the cylinder. A cat is pulling on one end of the rope with a force F while 10 mice can just barely prevent it from sliding by applying a total force .

a) Does the minimum force necessary to prevent the rope from sliding depend on the diameter of the cylinder?

b) Through what minimum angle about the cylinder should one mouse wrap the rope in order to prevent the cat from winning the game of tug of war?

Penalty (NGU)

How many revolutions can do the football at the penalty kick?