This set of Physics Multiple Choice Questions & Answers (MCQs) focuses on “Theorems of Perpendicular and Parallel Axes”.
1. A planar body is lying in the xz plane. What is the relation between its moment of inertia along the x, y & z axes?
a) Iz = Ix + Iy
b) Ix = Ix + Iz
c) Iy = Ix + Iz
d) Iz = Ix = Iy, because body is planar
Explanation: The body is lying in the xz plane. The perpendicular axis theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of moment of inertia about two perpendicular axes in the plane of the body. Iy is perpendicular to plane of body, so Iy = Ix + Iz.
2. Perpendicular axis theorem can be applied for which of the following bodies?
a) Ring having radius R & negligible cross section
b) Disc of radius R and thickness t
c) Cylinder of radius R and height h
d) A cube of side ‘a’
Explanation: Perpendicular axis theorem can only be applied to planar bodies. From the given options only the ring is planar.
3. Consider two perpendicular axis in the plane of a planar body, such that I1 = 2 I2. The moment of inertia about an axis perpendicular to the plane and passing through intersection of I1 & I2 is 9kgm2. Find the value of I1& I2.
a) I1 = 9kg m2, I2 = 4.5kgm2
b) I1 = 3kg m2, I2 = 6kg m2
c) I1 = 6kg m2, I2 = 3kg m2
d) I1 = 18kg m2, I2 = 9kg m2
Explanation: By perpendicular axis theorem: 9 = I1 + I2 = 3 I2.
∴ I2 = 3kgm2 & I1 = 2*3 = 6kgm2.
4. The moment of inertia of a planar disc about a diameter is 8kgm2. What is the moment of inertia about an axis passing through its centre and perpendicular to the plane of disc?
Explanation: The moment of inertia about any diametrical axis will be the same. So, we can consider two perpendicular diameters and use the perpendicular axis theorem to get the moment of inertia about the axis which is perpendicular to the plane.
Thus, moment of inertia = 8+8
5. Let I1 be the moment of inertia about the centre of mass of a thick asymmetrical body. Let I2 be the moment of inertia about an axis parallel to I1. The distance between the two axes is ‘a’ & the mass of the body is ‘m’. Find the relation between I1 & I2.
a) I2 = I1 – ma2
b) I1 = I2 – ma2
c) I2 = I1
d) Parallel axis theorem can’t be used for a thick asymmetrical body
Explanation: Parallel axis theorem can be used for any body. The parallel axis theorem states that moment of inertia about an axis perpendicular to an axis passing through centre of mass is given by:
I = ICOM + ma2, where m is mass of the body & ‘a’ is the distance between the axes. So, I2= I1 + ma2 OR I1 = I2 – ma2.
6. What is the moment of inertia of a rod, of mass 1kg & length 6m, about an axis perpendicular to rod’s length and at a distance of 1.5m from one end?
Explanation: Moment of inertia about an axis perpendicular to length and passing through COM is equal to MI2/12. To find I about given axis, we use parallel axis theorem,
I = ICOM + ma2, where a is the distance between the axis and m is mass of the body.
I = MI2/12 + Ma2
= 1*36/12 + 1*2.25
= 3 + 2.25 = 5.25kgm2.
7. The moment of inertia of a ring about a tangent is 4kgm2. What is the moment of inertia about an axis passing through the centre of the ring and perpendicular to its plane? Mass of the ring is 2kg & diameter is 2m.
Explanation: Using parallel axis theorem we can find I about centre of mass.
∴ I = 4 – 2*1 = 2kgm2.
Now using perpendicular axis theorem we get I1 about the desired axis.
∴ I1 = I + I = 4kgm2.
8. The moment of inertia of a planar square about a planar axis parallel to one side is 10kgm2. What is the moment of inertia about a diagonal?
Explanation: Using perpendicular axis theorem we can say that sum of moment of inertia about two perpendicular axes will be the same as sum of moment of inertia about other 2 perpendicular axes. Let I1 be the moment of inertia about a diagonal, we can say:
I1 + I1= 10 + 10 = 20kgm2
Or I1 =10kgm2.
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