four masses of 2 kg,4kg,3kg, and 6 kg are kept at the corner of square ABCD are side 1 m then (A) find moment of inertia about an axis which passes through a point of intersection its diagonal and perpendicular on it .
(b) about the axis AB. (C) about diagonal
To find the moment of inertia for each scenario, we would need to use the basic formula for moment of inertia:
(a) Moment of inertia about an axis passing through a point of intersection of the diagonal and perpendicular on it:
1. Draw the square ABCD and label the masses at the corners.
2. The diagonal of the square passing through the point of intersection is a line connecting opposite corners, such as AC or BD.
3. The perpendicular line on the diagonal is a line connecting the midpoint of the diagonal to the opposite side of the square, such as CE or BF, where E and F are midpoints of AC and BD respectively.
4. Since the masses are point masses, we can calculate the moment of inertia for each mass separately and then add them up.
The formula for moment of inertia of a point mass is: I = mr^2, where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.
For the mass at corner A:
- The distance from the axis passing through the point of intersection to mass A is the length of the diagonal, which is sqrt(1^2 + 1^2) = sqrt(2).
- The moment of inertia for mass A is I_A = m_A * r_A^2 = 2kg * (sqrt(2))^2 = 4kgm^2.
Repeat this process for the other three masses and add up the moment of inertia for each mass to get the total moment of inertia about the given axis.
(b) Moment of inertia about the axis AB:
1. The axis AB is a side of the square, which is 1m in length.
2. Again, use the formula I = mr^2 for each mass and calculate the moment of inertia individually.
For the mass at corner A:
- The distance from the axis AB to mass A is half the distance of AB, which is 1m/2 = 0.5m.
- The moment of inertia for mass A is I_A = m_A * r_A^2 = 2kg * (0.5)^2 = 0.5kgm^2.
Repeat this process for the other three masses and add up the moment of inertia for each mass to get the total moment of inertia about the given axis.
(c) Moment of inertia about the diagonal:
1. The diagonal of the square is sqrt(1^2 + 1^2) = sqrt(2) in length.
2. Use the formula I = mr^2 for each mass and calculate the moment of inertia individually.
For the mass at corner A:
- The distance from the diagonal to mass A is half the diagonal length, which is sqrt(2)/2.
- The moment of inertia for mass A is I_A = m_A * r_A^2 = 2kg * (sqrt(2)/2)^2 = kgm^2.
Repeat this process for the other three masses and add up the moment of inertia for each mass to get the total moment of inertia about the given diagonal axis.
To find the moment of inertia of a system of masses about a given axis, we need to consider the individual contributions of each mass.
Let's start by labeling the masses at the corners of the square ABCD:
A - 2 kg
B - 4 kg
C - 3 kg
D - 6 kg
(a) Moment of inertia about an axis passing through a point of intersection of its diagonal and perpendicular on it:
1. To find the moment of inertia about this axis, we need to calculate the distance of each mass from the axis.
2. The axis passes through the intersection point of the square's diagonal and the perpendicular, which is also the square's center. Let's call this point O.
3. The distance of mass A from the axis is the diagonal length AO. Since ABCD is a square with side length 1 m, AO is the distance between A and O.
Since AO is also the hypotenuse of a right-angled triangle with legs of length 1 m, we can use the Pythagorean theorem to find AO. AO = sqrt(1^2 + 1^2) = sqrt(2) m.
4. Similarly, the distances of masses B, C, and D from the axis are also sqrt(2) m.
5. The moment of inertia of each mass about the given axis is given by its mass multiplied by the square of its distance from the axis.
Thus, the moment of inertia of mass A = 2 kg * (sqrt(2) m)^2 = 4 kg * 2 = 8 kg.m^2.
The moment of inertia of mass B = 4 kg * (sqrt(2) m)^2 = 4 kg * 2 = 8 kg.m^2.
The moment of inertia of mass C = 3 kg * (sqrt(2) m)^2 = 3 kg * 2 = 6 kg.m^2.
The moment of inertia of mass D = 6 kg * (sqrt(2) m)^2 = 6 kg * 2 = 12 kg.m^2.
6. The total moment of inertia about the given axis is the sum of the individual moments of inertia.
Total moment of inertia = 8 kg.m^2 + 8 kg.m^2 + 6 kg.m^2 + 12 kg.m^2 = 34 kg.m^2.
(b) Moment of inertia about the axis AB:
1. To find the moment of inertia about this axis, we need to calculate the distance of each mass from the axis AB.
2. Axis AB passes through the corner points A and B. The distance of mass A from the axis AB is the perpendicular distance from point A to axis AB.
Since ABCD is a square with side length 1 m, this distance is simply 1 m.
3. Similarly, the distance of mass B from the axis AB is also 1 m.
4. The distances of masses C and D from the axis AB are the diagonals of the square.
The diagonals of a square are given by d = sqrt(2) * side length.
Thus, the distance of masses C and D from the axis AB = sqrt(2) * 1 m = sqrt(2) m.
5. The moment of inertia of each mass about the axis AB is given by its mass multiplied by the square of its distance from the axis.
Thus, the moment of inertia of mass A = 2 kg * (1 m)^2 = 2 kg.m^2.
The moment of inertia of mass B = 4 kg * (1 m)^2 = 4 kg.m^2.
The moment of inertia of masses C and D = 3 kg * (sqrt(2) m)^2 = 3 kg * 2 = 6 kg.m^2 (for both masses).
6. The total moment of inertia about the axis AB is the sum of the individual moments of inertia.
Total moment of inertia = 2 kg.m^2 + 4 kg.m^2 + 6 kg.m^2 + 6 kg.m^2 = 18 kg.m^2.
(c) Moment of inertia about the diagonal:
1. To find the moment of inertia about the diagonal, we consider the axis passing through the intersection points of the square's diagonals.
Let's call this axis OD.
2. Axis OD is a diagonal of the square, so its length is sqrt(2) * side length = sqrt(2) m.
3. The distances of masses A and C from the axis OD are the perpendicular distances from points A and C to the axis OD.
Since ABCD is a square with side length 1 m, these distances are both 1 m.
4. Similarly, the distances of masses B and D from the axis OD are also 1 m.
5. The moment of inertia of each mass about the axis OD is given by its mass multiplied by the square of its distance from the axis.
Thus, the moment of inertia of mass A = 2 kg * (1 m)^2 = 2 kg.m^2.
The moment of inertia of mass B = 4 kg * (1 m)^2 = 4 kg.m^2.
The moment of inertia of mass C = 3 kg * (1 m)^2 = 3 kg.m^2.
The moment of inertia of mass D = 6 kg * (1 m)^2 = 6 kg.m^2.
6. The total moment of inertia about the axis OD is the sum of the individual moments of inertia.
Total moment of inertia = 2 kg.m^2 + 4 kg.m^2 + 3 kg.m^2 + 6 kg.m^2 = 15 kg.m^2.
Therefore, the moment of inertia for the given masses is:
(a) about the axis passing through the point of intersection of the diagonal and the perpendicular on it: 34 kg.m^2.
(b) about the axis AB: 18 kg.m^2.
(c) about the diagonal: 15 kg.m^2.