A weight with mass mw=150 gmw=150 g is tied to a piece of thread wrapped around a spool, which is suspended in such a way that it can rotate freely. When the weight is released, it accelerates toward the floor as the thread unwinds. Assume that the spool can be treated as a uniform solid cylinder of radius R=4.00 cmR=4.00 cm and mass s=200 gMs=200 g . Find the magnitude aa of the acceleration of the weight as it descends. Assume the thread has negligible mass and does not slip or stretch as it unwinds.

The acceleration is  

Explanation:

From the question we are told that

  The mass of the weight is  

    The radius of the spool is  

     The mass of the spool is

Generally the net force acting on the weight is mathematically represented as

Here W is the weight of the weight which is mathematically represented as

and  T  is the tension on the thread

So

Generally this net force acting on the weight can be mathematically represented as  

Here is the a is the acceleration of the system (i.e acceleration of the weight as a result of its weight and the tension on the rope )

    So

Generally the torque which the spool experiences can be mathematically represented as

This torque is also mathematically represented as

Here is moment of inertia of the spool which is mathematically represented as

while   is the angular acceleration of the spool which is mathematically represented as

so

=>  

So

=>  

Now substituting this formula for T  into the equation above  

=>    

=>    

=>    

probably not if net force is zero

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .