250+ TOP MCQs on Theorems of Perpendicular and Parallel Axes | Class 11 Physics

Physics Multiple Choice Questions 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) I_z = I_x + I_y
b) I_x = I_x + I_z
c) I_y = I_x + I_z
d) I_z = I_x = I_y, because body is planar
Answer: c
Clarification: 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. I_y is perpendicular to plane of body, so I_y = I_x + I_z.

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’
Answer: a
Clarification: 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
Answer: c
Clarification: 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?
a) 8kgm²
b) 16kgm²
c) 4kgm²
d) 2√2kgm²
Answer: b
Clarification: 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
= 16kgm².

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
Answer: b
Clarification: 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 = I_COM + 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?
a) 0.75kgm²
b) 3kgm²
c) 5.25kgm²
d) 14.25kgm²
Answer: c
Clarification: 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 = I_COM + 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.
a) 2kgm²
b) 4kgm²
c) 8kgm²
d) 1kgm²
Answer: b
Clarification: 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?
a) 10kgm²
b) 5kgm²
c) 20kgm²
d) 1kgm²
Answer: a
Clarification: 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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