Three objects lie in the x, y plane. Each rotates about the...

Three objects lie in the $x, y$ plane. Each rotates about the $z$ axis with an angular speed of $6.00 \mathrm{rad} / \mathrm{s} .$ The mass $m$ of each object and its perpendicular distance $r$ from the $z$ axis are as follows: $(1) m_{1}=6.00 k g$ and $r_{1}=2.00 \mathrm{~m},(2)$ $m_{2}=4.00 \mathrm{~kg}$ and $r_{2}=1.50 \mathrm{~m},(3) \mathrm{m}_{3}=3.00 \mathrm{~kg}$ and $r_{3}=3.00 \mathrm{~m} .$ (a) Find the tangential speed of each object. (b) Determine the total kinetic energy of this system using the expression $\mathrm{KE}=\frac{1}{2} m_{1} v_{1}^{2}+\frac{1}{2} m_{2} v_{2}^{2}+\frac{1}{2} m_{3} v_{3}^{2}$ (c) Obtain the moment of inertia of the system. (d) Find the rotational kinetic energy of the system using the relation $\frac{1}{2} I \omega^{2}$ to verify that the answer is the same as that in (b).

Three objects lie in the $x, y$ plane. Each rotates about the $z$ axis with an angular speed of $6.00 \mathrm{rad} / \mathrm{s} .$ The mass $m$ of each object and its perpendicular distance $r$ from the $z$ axis are as follows: $(1) m_{1}=6.00 k g$ and $r_{1}=2.00 \mathrm{~m},(2)$ $m_{2}=4.00 \mathrm{~kg}$ and $r_{2}=1.50 \mathrm{~m},(3) \mathrm{m}_{3}=3.00 \mathrm{~kg}$ and $r_{3}=3.00 \mathrm{~m} .$ (a) Find the tangential
speed of each object. (b) Determine the total kinetic energy of this system using the expression $\mathrm{KE}=\frac{1}{2} m_{1} v_{1}^{2}+\frac{1}{2} m_{2} v_{2}^{2}+\frac{1}{2} m_{3} v_{3}^{2}$ (c) Obtain the moment of inertia of the system. (d) Find the rotational kinetic energy of the system using the relation $\frac{1}{2} I \omega^{2}$ to verify that the answer is the same as that in (b).

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy represents the energy due to the rotational motion of a system and is given by 1/2 I ?^2. This expression provides an alternative calculation for the kinetic energy of a rotating system, linking the moment of inertia to the angular speed.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion around an axis and is calculated, for point masses, as I = m r^2. It plays a crucial role in determining how the mass is distributed relative to the axis of rotation.

Tangential Speed

Tangential speed is the linear speed of a point at a given distance from the axis of rotation, calculated by multiplying the angular speed (?) by the radial distance (r). This concept relates the circular motion of an object to its equivalent linear motion along the circumference.

Angular Speed

Angular speed is the rate at which an object rotates around an axis, measured in radians per second. It defines the steady rate of rotation for all objects in a system that are rotating about the same axis.

Kinetic Energy in Rotational Motion

Kinetic energy in rotational motion for point masses is often computed using the formula KE = 1/2 m v^2, where v is the tangential speed. This form of energy quantifies the energy associated with the linear motion derived from an object’s rotation.

Transcript

00:01 For number 45, we're given these three objects, and they're rotating around this axis in the z, the z axis.

00:09 So this red dot, imagine that is sticking straight out of the page, and then these are all going around it.

00:17 And i think it's easier to just draw those.

00:20 So you can see the mass of every one of them, and you can see the distance, the radius for each one of them.

00:25 And they all have the same speed, the same angular speed, six radiance per se.

00:30 Okay, first of all we're supposed to find the tangential speed or the linear speed for each one of those.

00:36 So that's part a.

00:39 And remember how the linear rotational related.

00:45 The rotational speed is the linear speed divided by the radius.

00:50 So i'm just going to do this for each one of them.

00:53 So they all have an angular speed of six.

01:00 I'll go across this way, i guess.

01:04 So this is, the first one's going to be six.

01:06 The velocity of mass 1, the radius of mass 1 is 2.

01:12 So you can tell that velocity 1 is going to be 12.

01:21 And then for the next one, still 6.

01:25 Looking for the linear velocity of 2, and this one has a distance of 1 .5.

01:31 So just 1 .5 times 6.

01:33 9.

01:42 3rd mass, third velocity, third radius is 3 meters.

01:51 So this is going to have a linear velocity of 18.

01:59 Part b ask for the kinetic energy, the total kinetic energy, so i'm finding the kinetic energy of each one of these, and then i'll just add them.

02:12 So remember the equation for kinetic energy is one -half mv squared, and it's a linear velocity i just found that.

02:32 So for the first one, one -half, i can get from my picture that my mass is six, i just calculated my velocity was 12, that gets squared, so i get 432, that'll be joules...

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