when temperature increases from t to # t and the moment of inertia is increase from I to #I. the coefficent of linear expansion is alpha .THE RATIO OF #I\I IS
To find the ratio of #I to I when the temperature increases from t to #t, and the coefficient of linear expansion is alpha, we can use the formula for the moment of inertia of a homogeneous object:
I = k * M * R^2
Where:
I is the moment of inertia
M is the mass of the object
R is the radius of gyration (which depends on the shape of the object)
k is a constant
Assuming the mass and shape of the object remain constant, the moment of inertia can be calculated as:
I = k * R^2
Now, let's consider the change in the moment of inertia when the temperature increases from t to #t. We are given that the moment of inertia increases from I to #I.
Therefore, the ratio #I/I can be calculated as:
#I/I = (#I) / I
To find (#I) / I, we need to find the change in the moment of inertia, #I.
The change in moment of inertia is given by:
#I = I2 - I1
where I2 is the moment of inertia at #t and I1 is the moment of inertia at t.
Now, we need to find the expression for I2 and I1 in terms of R.
The coefficient of linear expansion, alpha, relates the change in length (delta L) of an object to the change in temperature (delta T) through the formula:
delta L = alpha * L * delta T
In our case, the radius of gyration, R, can be considered as the "length" of the object. So, we can express the change in radius of gyration as:
delta R = alpha * R * delta T
Now, let's substitute the values of R and #R (change in R) into the expression for the moment of inertia:
I2 = k * (#R)^2
I1 = k * R^2
Substituting these expressions into the equation for #I, we get:
#I = I2 - I1
= (k * (#R)^2) - (k * R^2)
= k * [(#R)^2 - R^2]
Finally, we can calculate the ratio of #I/I:
#I/I = (#I) / I
= [k * ((#R)^2 - R^2)] / (k * R^2)
= (#R^2 - R^2) / R^2
Simplifying this expression, we find:
#I/I = (#R^2 - R^2) / R^2
Therefore, the ratio of #I to I is (#R^2 - R^2) / R^2.