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) $m_{1}=6.00 \mathrm{kg}$ and $r_{1}=$ $2.00 \mathrm{m},(2) m_{2}=4.00 \mathrm{kg}$ and $r_{2}=1.50 \mathrm{m},(3) m_{3}=3.00 \mathrm{kg}$ 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 $\mathrm{KE}_{\mathrm{R}}=\frac{1}{2} \mathrm{I} \omega^{2}$ to verify that the answer is the same as the answer to (b).

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates about an axis. It is expressed in radians per second and serves as a fundamental parameter in rotational dynamics, linking the motion of rotating objects to their linear and angular quantities.

Tangential Speed

Tangential speed is the linear speed of a point located a certain distance from the axis of rotation. It is directly proportional to both the angular speed and the radius of rotation, given by the relation v = ?r, which connects rotational motion to linear motion.

Kinetic Energy

Kinetic energy in the context of motion is the energy an object possesses due to its speed. For moving objects, especially in rotational systems when considering individual points, the translational kinetic energy is given by KE = 1/2 mv², where m is mass and v is the tangential speed.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to angular acceleration by highlighting how its mass is distributed relative to the axis of rotation. For point masses, it is computed as I = mr², summing the contributions from all masses in a system to characterize the overall rotational inertia.

Rotational Kinetic Energy

Rotational kinetic energy quantifies the energy associated with the rotation of an object and is given by KE_rot = 1/2 I ?². This formulation links the rotational energy directly to both the moment of inertia and the square of the angular speed, underscoring the role these quantities play in rotational dynamics.

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