Posted by sandhu on Tuesday, November 30, 2010 at 1:17am.


A T is made of two identical 740 g thin solid rods. The moment of inertia of the T about an axis perpendicular to the screen through point at one end of a rod is determined experimentally to be 0.68 kg-m2. What is the length of the rods?

physics - bobpursley, Tuesday, November 30, 2010 at 9:33am
I have no idea where the screen is. But I suspect you need the parallel axis theorem.

physics - drwls, Tuesday, November 30, 2010 at 9:41am
The answer depends upon which end of which rod is the axis of rotation. You have not provided that information. Is it there the rods intersect? Or the bottom of the "T" ?

physics - sandhu, Tuesday, November 30, 2010 at 7:27pm
the bottom of the T or at the top of an inverted T

To solve this problem, we will need to use the parallel axis theorem. The moment of inertia of an object about an axis parallel to and a distance "h" away from an axis passing through its center of mass is given by the equation:

I = Icm + Mh^2

Where I is the moment of inertia about the parallel axis, Icm is the moment of inertia about the center of mass axis, M is the mass of the object, and h is the distance between the two axes.

In this case, we are given that the moment of inertia about the perpendicular axis passing through one end of the rod is 0.68 kg-m^2. Let's assume that the length of each rod is "L".

For a thin rod, the moment of inertia about an axis perpendicular to the length of the rod and passing through one end is given by the equation:

I = (1/3)ML^2

Since the two rods are identical, their masses are the same. Let's denote the mass of each rod as "m". Therefore, the moment of inertia about the perpendicular axis through one end of each rod is:

Icm = (1/3)mL^2

To find the moment of inertia about the perpendicular axis passing through the bottom of the "T" (or the top of an inverted "T"), we need to apply the parallel axis theorem. Let's denote the distance between these two axes as "h".

Using the parallel axis theorem, we can write the equation:

I = Icm + Mh^2

Substituting the given values, we have:

0.68 kg-m^2 = (1/3)mL^2 + 2mh^2

Since two rods are used, we have a total mass of 2m.

Now, we need to find the relationship between "h" and "L". From the diagram, we can see that "h" is equal to half the length of each rod. Therefore, h = L/2.

Substituting this value into the equation, we have:

0.68 kg-m^2 = (1/3)mL^2 + 2m(L/2)^2

Simplifying, we get:

0.68 kg-m^2 = (1/3)mL^2 + (1/2)mL^2

Combining the terms on the right side, we have:

0.68 kg-m^2 = (5/6)mL^2

To simplify further, we can multiply both sides of the equation by 6/5:

0.68 kg-m^2 * (6/5) = mL^2

Multiplying the values, we get:

(0.408 kg-m^2) = mL^2

Now, we can divide both sides of the equation by "m" to isolate "L^2":

L^2 = (0.408 kg-m^2)/m

Finally, we take the square root of both sides to find the length of each rod "L":

L = √((0.408 kg-m^2)/m)

Therefore, the length of each rod is given by this equation.

To solve this problem, we need to use the parallel axis theorem. The moment of inertia of an object about an axis parallel to and a distance "d" away from the object's center of mass is given by the sum of its moment of inertia about its center of mass and the product of its mass and the square of the distance "d."

In this case, we are given the moment of inertia of the T (0.68 kg-m^2) and the mass of each rod (740 g).

Let's assume the length of each rod is "L" and the distance from the end of the rod to the axis of rotation is "d."

The moment of inertia of each rod about its center of mass is given by (1/12)mL^2, where "m" is the mass of each rod.

Using the parallel axis theorem, the total moment of inertia of the T is the sum of the moment of inertia of each rod about its center of mass and the additional moment of inertia due to the distance "d" from the center of mass to the axis of rotation.

Mathematically, it can be written as:

I_total = (1/12)mL^2 + (m)(d^2)

Since both rods of the T are identical, we can simplify the equation to:

2*((1/12)mL^2) + (m)(d^2) = 0.68 kg-m^2

Now, let's substitute the given values. The mass of each rod is 740 g, which is equal to 0.74 kg. And let's also assume the distance "d" is equal to the length of the rod "L".

Substituting these values into the equation, we get:

2*((1/12)*(0.74)*(L^2)) + (0.74)*(L^2) = 0.68

Simplifying further, we get:

((1/6)*(0.74)*(L^2)) + (0.74)*(L^2) = 0.68

Multiplying both sides by 6 to eliminate the fraction, we get:

(0.74)*(L^2) + 6*(0.74)*(L^2) = 4.08

Simplifying further, we get:

7.44*(L^2) = 4.08

Dividing both sides by 7.44, we get:

L^2 = 0.5484

Taking the square root of both sides, we get:

L = √0.5484

L ≈ 0.74 meters

Therefore, the length of each rod in the T is approximately 0.74 meters.