A uniform cylinder of radius 14 cm and mass 14 kg is mounted so as to rotate fre

Question

A uniform cylinder of radius 14 cm and mass 14 kg is mounted so as to rotate freely about a horizontal axis that is parallel to and 6.7 cm from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position? --------------------------------------------------------------------------------------------- Four identical particles of mass 0.858 kg each are placed at the vertices of a 3.60 m x 3.60 m square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

Explanation / Answer

mass , m = 14 Kg

radius ,r = 0.14 m

distance of axis , d = 6.7 cm

d = 0.067 m

A) Using parallel axis theorum

moment of inertia of cyclinder about the axis of rotation = 0.5 * m * r^2 + m * d^2

moment of inertia of cyclinder about the axis of rotation = 14 * ( 0.5 * 0.14^2 + 0.067^2 )

moment of inertia of cyclinder about the axis of rotation = 0.2 Kg.m^2

the moment of inertia of cyclinder about the axis of rotation is 0.2 Kg.m^2

b)

h = 0.067 m

Using conseration of energy

m * g * h = 0.5 * I * w^2

14 * 9.8 * 0.067 = 0.5 * 0.2 * w^2

solving for w

w = 9.59 rad/s

the angular speed of the cyclinder is 9.59 rad/s

----------------------------------------

mass , m= 0.858 Kg

d = 3.6 m

a) for the moment of inertia about the axis passing through the midpoint

moment of inertia passing through the midpoint = 4 * m * (d/2)^2

moment of inertia passing through the midpoint = 4 * 0.858 * (3.6/2)^2

moment of inertia passing through the midpoint = 11.2 Kg.m^2

the moment of inertia passing through the midpoint is 11.2 Kg.m^2

b)

for the perpendicular axis

moment of inertia = 2 * m * (d/2)^2 + 2 * m * ((d/2)^2 + d^2)

moment of inertia = 2 * 14 * (1.8)^2 + 2 * 14 * (1.8^2 + 3.6^2)

moment of inertia = 33.36 Kg.m^2

the moment of inertia is 33.36 Kg.m^2

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