I have a problem in mechanical physics where I\'m given the linear mass density

Question

I have a problem in mechanical physics where I'm given the linear mass density of a rigid plastic material. Here's the problem for context: A 42 cm diameter wheel, consisting of a rim and six spokes, is constructed from a thin, rigid plastic material having a linear mass denisty of 25 g/cm. The wheel is released from rest at the top of a hill 58 m high. Find the velocity at the bottom of the hill.

What is linear mass density??? I noticed in another problem it is equal to 2piRlambda in some cases, while only lambdaR in others. Where do these equations even come from??

Explanation / Answer

Linear mass density is the measure of a quantity of any characteristic value per unit of length. If I am given a rod of linear mass density lambda, I can get the mass of a certain length of the rod simply by multiplying lambda with length of the portion whose mass I am interested in.

So, linear mass density of an object is simply:

lambda = mass of the object / total length of the object

If the object is circular in shape(wheel), you need to multiply its whole length i.e. circumference with lambda to get the mass of the whole circular object(wheel), hence 2piRlambda. If you need the mass of a straight object(spoke of wheel), you need to mutiply its whole length with lambda, hence lambdaR ( for spoke of wheel of radius R)

I hope that clears it up. :)

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