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

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.

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.

find the surface area of the sphere to the nearest square unit. use a calculator.

a. 254 in.2

b. 64 in.2

c. 127 in.2

d. 1,018 in.2