The moment of inertia is a property of an object that resists a change in angular acceleration. It plays the same role for rotation as mass plays for linear motion. For example, if we have two objects of mass m₁ and m₂ (m₁ > m₂) experiencing a common force F then by Newton's second law the acceleration of each object is a₁ = F/m₁ and a₂ = F/m₂. Since the first mass is larger than the second then the first object accelerates more slowly. The equivalent Newton's law in rotation is T = Ia where I is the moment of inertia, a the angular acceleration, and 7 the torque. By analogy to linear motion, if two objects experiences a common torque then the object with the higher moment of inertia will have a slower angular acceleration. The moment of inertia for a set of objects of mass m, rotating about a common axis is defined as I=mr²
The moment of inertia is a property of an object that resists a change in angular acceleration. It plays the same role for rotation as mass plays for linear motion. For example, if we have two objects of mass m₁ and m₂ (m₁ > m₂) experiencing a common force F then by Newton's second law the acceleration of each object is a₁ = F/m₁ and a₂ = F/m₂. Since the first mass is larger than the second then the first object accelerates more slowly. The equivalent Newton's law in rotation is T = Ia where I is the moment of inertia, a the angular acceleration, and 7 the torque. By analogy to linear motion, if two objects experiences a common torque then the object with the higher moment of inertia will have a slower angular acceleration. The moment of inertia for a set of objects of mass m, rotating about a common axis is defined as I=mr²
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Transcribed Image Text:1. The moment of inertia is a property of an object that resists a change in angular acceleration.
It plays the same role for rotation as mass plays for linear motion. For example, if we have
two objects of mass m₁ and m₂ (m₁ > m₂) experiencing a common force F then by Newton's
second law the acceleration of each object is a₁ = F/m₁ and 2 F/m₂. Since the first
mass is larger than the second then the first object accelerates more slowly. The equivalent
Newton's law in rotation is
=
T = Ia
where I is the moment of inertia, a the angular acceleration, and 7 the torque. By analogy
to linear motion, if two objects experiences a common torque then the object with the higher
moment of inertia will have a slower angular acceleration.
The moment of inertia for a set of objects of mass m, rotating about a common axis is
defined as
1=Σm₂r²
where r; is the distance of the ith object to the axis of rotation. If there are many particles
that make up a larger object then this sum transforms into an integral,
I - JJJ pr² av,
where p is the mass density and V the volume of the object.
In this exercise we will explore moment of inertia by rolling two objects down an incline
plane in the Experimental Math Lab Space.
Consider a cylindrical shell of constant density p, inner radius 7₁, outer radius R, and height
H. Assume that the cylinder is oriented along the z-axis.
(a) Find the mass, m, of the cylinder.
(b) Show that the z-axis goes through the centre of mass of the cylinder
(c) Assume that the cylinder rotates around the z-axis. Find the moment of inertia of the
cylinder in terms of the mass m. Why does the height of the cylinder not matter?
(d) Compute the moment of inertia from (1c) for the specific cases of a solid cylinder (₁ = 0)
and a thin cylindrical shell (r₁ → R). If the objects race down an inclined plane, which
one reaches the bottom first?
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