Question

The rigid object shown is rotated about an
axis perpendicular to the paper and through
center point O. The total kinetic energy of
the object as it rotates is 8 J.

a) What is the moment of inertia of the ob-
ject? Neglect the mass of the connecting rods
and treat the masses as point masses.
Answer in units of kg · m2

b)What is the angular velocity of the object?
Answer in units of rad/s

Answer 1

Answer:

Since the object is rotating, it will have a Rotational Kinetic Energy and Angular Velocity, that can be related by the equation below:

K.E= ½Iω2.

Defined as

Rotational kinetic energy = ½ moment of inertia * (angular speed)2..

K.E = 8J.

Therefore, making the moment of inertia, I the subject of the relation, we have

2*K.E = I * (Angular Velocity)2

Divide both sides by (Angular Velocity)2 and putting K.E = 8J,

(a) I = (2 x 8)/(Angular Velocity)2

I = 16/(ω2) Kg.m2

(b) Angular Velocity ω is calculated by making ω the subject of the relation

ω2 = (2 *K.E)/I

ω2 = (2 x 8)I = 16/I

Taking square root of both sides

ω = Sqrt(16/I) = 4/Sqrt(I) rad/s

Explanation:

When an object is rotating about its center of mass, its rotational kinetic energy is K = ½Iω2.

Rotational kinetic energy = ½ moment of inertia * (angular speed)2.

When the angular velocity of a spinning wheel doubles, its kinetic energy increases by a factor of four.

When an object has translational as well as rotational motion, we can look at the motion of the center of mass and the motion about the center of mass separately. The total kinetic energy is the sum of the translational kinetic energy of the center of mass (CM) and the rotational kinetic energy about the CM, Which is center O in the question.

Answer 2

(A). The moment of inertia of the object is = 16/(ω2) Kg.m2

(B.) The angular velocity of the object is = Sqrt(16/I) = 4/Sqrt(I) rad/s

What is Kinetic Energy?

When the object is rotating, it will have a Rotational Kinetic Energy and Angular Velocity, that can be related by the equation below:

K.E= ½Iω2.

Described as Rotational kinetic energy is = ½ moment of inertia * (angular speed)2.

K.E = 8J.

Thus, assembling the moment of inertia, 'I' the subject of the relation, we have

2*K.E is = I * (Angular Velocity)2

Divide both sides by (Angular Velocity)2 and putting K.E is = 8J,

(a) I = (2 x 8)/(Angular Velocity)2

Therefore, I = 16/(ω2) Kg.m2

(b) The Angular Velocity ω is calculated by making ω the subject of the relation

ω2 is = (2 *K.E)/I

ω2 is = (2 x 8)I = 16/I

Then, we Taking square root of both sides

Therefore, ω = Sqrt(16/I) = 4/Sqrt(I) rad/s

When an object is rotating around its center of mass, its rotational kinetic energy is K = ½Iω2.

Rotational kinetic energy is = ½ moment of inertia * (angular speed)2.

When the angular velocity of a spinning rotation double, its kinetic energy increases by a factor of four.

When an object has translational as well as rotational motion, we can glance at the motion of the center of mass and the motion regarding the center of mass individually.

When The total kinetic energy is the totality of the translational kinetic energy of the center of mass (CM) and the rotational kinetic energy regarding the CM, Which is center O in the query.

Find more information about Kinetic Energy here:

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