Review. As shown in Figure P10.71, two blocks are connected...

Review. As shown in Figure P10.71, two blocks are connected by a string of negligible mass passing over a pulley of radius $r=0.250 \mathrm{m}$ and moment of inertia $I .$ The block on the frictionless incline is moving with a constant acceleration of magnitude $a=2.00 \mathrm{m} / \mathrm{s}^{2} .$ From this information, we wish to find the moment of inertia of the pulley. (a) What analysis model is appropriate for the blocks? (b) What analysis model is appropriate for the pulley? (c) From the analysis model in part (a), find the tension $T_{1} \cdot(\mathrm{d})$ Similarly, find the tension $T_{2} .$ (e) From the analysis model in part (b), find a symbolic expression for the moment of inertia of the pulley in terms of the tensions $T_{1}$ and $T_{2},$ the pulley radius $r,$ and the acceleration $a .(\mathrm{f})$ Find the numerical value of the moment of inertia of the pulley.

Review. As shown in Figure P10.71, two blocks are connected by a string of negligible mass passing over a pulley of radius $r=0.250 \mathrm{m}$ and moment of inertia $I .$ The block on the frictionless incline is moving with a constant acceleration of magnitude $a=2.00 \mathrm{m} / \mathrm{s}^{2} .$ From this information, we wish to find the moment of inertia of the pulley.
(a) What analysis model is appropriate for the blocks?
(b) What analysis model is appropriate for the pulley?
(c) From the analysis model in part (a), find the tension
$T_{1} \cdot(\mathrm{d})$ Similarly, find the tension $T_{2} .$ (e) From the analysis model in part (b), find a symbolic expression for the moment of inertia of the pulley in terms of the tensions $T_{1}$ and $T_{2},$ the pulley radius $r,$ and the acceleration $a .(\mathrm{f})$ Find the numerical value of the moment of inertia of the pulley.

Key Concepts

Free-Body Diagrams

Free-body diagrams are a fundamental tool in mechanics that allow one to isolate each object and identify all the forces acting upon it. They are crucial for setting up the correct equations of motion for both translational and rotational dynamics, especially in systems with multiple objects and interactions such as blocks and pulleys.

Newton’s Second Law for Translation

Newton's second law for translation (F = ma) is used to relate the net force acting on an object to its mass and acceleration. This concept is applied to analyze the linear motion of objects, such as blocks on an incline, enabling the determination of unknown forces like tensions when the objects are accelerating.

Newton’s Second Law for Rotation

Newton's second law for rotation (? = I?) extends the idea of force and acceleration to rotational motion. It relates the net torque acting on a rotating object to its moment of inertia and angular acceleration. This law is essential when examining how differences in string tensions create torques on a pulley, thereby affecting its rotational dynamics.

Tension in Strings

The concept of tension in strings pertains to the force transmitted along a rope or cable when it is pulled tight by forces acting from opposite ends. In ideal conditions, while a massless string generally exerts the same tension throughout, in systems with a rotating pulley, the tensions on either side may differ, which must be taken into account in the analysis of both linear and rotational motions.

Kinematic Constraint Relations

Kinematic constraint relations are used to link the translational motions of connected objects with the rotational motion of components such as pulleys. For instance, the condition that a string does not slip on a pulley provides a relationship between the linear acceleration of the blocks and the angular acceleration of the pulley, ensuring consistent motion analysis across the system.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on the distribution of mass relative to the axis of rotation and is a critical quantity when analyzing how applied torques, such as those resulting from differing tensions in a string, produce angular acceleration in a rotating body.

Transcript

00:01 In this question, you are given this setup.

00:02 Two blocks are connected by a string of net retrieval mass, passing over a pulley, and then the acceleration of the blocks is equal to 2 meters per second square.

00:15 And then there are five parts in this question and we want to find the moments of inertia of the pulley.

00:25 Okay, so in parts a, there are six parts, sorry, yeah.

00:33 Okay, so in part a, what analysis model is appropriate for the blocks.

00:38 We'll be using the particle under a net force model.

00:52 Then in part b, what analysis model is appropriate for the pulley.

00:57 It will be rigid object under a net torque model.

01:14 Okay, then in part c, using the model in part a, find a tension t1.

01:20 So using newton's second law okay so f net because to m a so i'm going to apply this equation along the incline okay so t1 minus m1g sine data is equal to m1a so t1 equals to m1 a plus g sine data okay putting in the numbers and one is 15 kg is 2 g is 9 .8 data is 37 degrees and then you get 118 newtons okay so this is the answer for part c then in parts d similarly look fine t2 so for t2 so for t2 so just look at the free body diagram for t2 so this is m2g and then t2 is pointing upward the acceleration in downward okay so using newton's second law okay fnats equals to m a m2a in this case so m2g minus t2 equals to m2 a the t2 equals to m2 g minus a this is equal to 20 9 point times 9 .8 minus 2 this is equal to 156 newton's.

03:18 Okay, so this is the answer for part b...

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