48 



FOECES ACTING ON A SINGLE PAETICLE 



IVcosa 



If slipping does not take place, the particle is in equilibrium 

 under its weight W and the reaction between it and the plane. 

 Thus the reaction must consist of a vertical force W. We can 



resolve this into components 

 W cos a and W sin a perpen- 

 dicular to and up the plane. 

 The former is the normal com- 

 ponent of the reaction, the 

 latter is the frictional com- 

 ponent. Thus in the notation 

 already used we have 

 R = W cos a, 



FIG. 21 



so that in this case we have 

 F = ft tan a. 



As a increases, F and F/R both increase until, when a reaches 

 the value e, F/R reaches its limiting value //<, or tan e, and after 

 this slipping takes place. 



EXAMPLES 



1. A mass of 100 pounds placed on a rough horizontal plane is on the point 

 of starting into motion when acted on horizontally by a force equal to the 

 weight of 100 pounds. Find the arfgle of friction. 



2. A body placed on an inclined plane which makes an angle of 30 with the 

 horizontal is just on the point of moving down the plane when acted on by a 

 horizontal force equal to the weight of the body. Find the coefficient of friction. 



3. A man capable of exerting a pull of 200 pounds tries to drag a mass of 

 700 pounds over a horizontal road (coefficient of friction -J-). To help him, the 

 chain from a crane is attached to the mass, the chain hanging vertically. How 

 much tension must there be in the chain before the man can move the block ? 



4. An insect tries to crawl up the inside of a hemispherical bowl of radius a. 

 How high can it get, if the coefficient of friction between its feet and the bowl is ? 



5. A man trying to push a block of stone over ice pushes horizontally and 

 finds that just as soon as the stone begins to move his feet begin to slip. Show 

 that if he pushes upwards on the stone he will get it along without difficulty, but 

 that if he pushes downwards he cannot possibly move it. 



6. A smooth pulley is placed at the edge of a horizontal plane. A string 

 passes over it, having at one end a weight w hanging freely, and at the other 





