You are the technical consultant for an action-adventure...

You are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 20.0 -m-tall building and land on the ground safely at a final vertical speed of $4.00 \mathrm{~m} / \mathrm{s}$. At the edge of the building's roof, there is a $100 .-\mathrm{kg}$ drum that is wound with a sufficiently long rope (of negligible mass), has a radius of $0.500 \mathrm{~m}$, and is free to rotate about its cylindrical axis with a moment of inertia $I_{0}$. The script calls for the 50.0 -kg stuntman to tie the rope around his waist and walk off the roof. a) Determine an expression for the stuntman's linear acceleration in terms of his mass $m$, the drum's radius $r$ and moment of inertia $I_{0}$. b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of $4.00 \mathrm{~m} / \mathrm{s},$ and use this value to calculate the moment of inertia of the drum about its axis. c) What is the angular acceleration of the drum? d) How many revolutions does the drum make during the fall?

You are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 20.0 -m-tall building and land on the ground safely at a final vertical speed of $4.00 \mathrm{~m} / \mathrm{s}$. At the edge of the building's roof, there is a $100 .-\mathrm{kg}$ drum that is wound with a sufficiently long rope (of negligible mass), has a radius of $0.500 \mathrm{~m}$, and is free to rotate about its cylindrical axis with a moment of inertia $I_{0}$. The script calls for the 50.0 -kg stuntman to tie the rope around his waist and walk off the roof.
a) Determine an expression for the stuntman's linear acceleration in terms of his mass $m$, the drum's radius $r$ and moment of inertia $I_{0}$.
b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of $4.00 \mathrm{~m} / \mathrm{s},$ and use this value to calculate the moment of inertia of the drum about its axis.
c) What is the angular acceleration of the drum?
d) How many revolutions does the drum make during the fall?

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about an axis. It depends not only on the mass of the object but also on how that mass is distributed relative to the axis of rotation. In problems involving rotating bodies, it serves as the rotational analogue of mass in linear motion and is crucial for calculating angular acceleration under applied torques.

Relationship between Linear and Angular Acceleration

The connection between the linear acceleration of an object and the angular acceleration of a rotating body is provided by the relation a = r?, where a is linear acceleration, r is the radius of the rotating object, and ? is angular acceleration. This relation is particularly important in systems where translational movement is linked to rotational motion, such as a rope unwinding from a drum.

Newton's Second Law for Translation

This principle describes how the net force acting on a body produces its acceleration. In the context of a falling object, it relates the gravitational force and any additional forces (like tension from the rope) to the object’s linear acceleration. It is a foundational concept in mechanics used to analyze the motion of masses subject to forces.

Rotational Dynamics

Rotational dynamics is the study of the motion of bodies as they rotate about an axis. It involves concepts such as torque, angular acceleration, and the distribution of mass, which is encapsulated in the moment of inertia. This framework is essential when dealing with objects that roll, spin, or rotate, and it applies when a linear motion is coupled with rotational motion.

Kinematic Equations for Constant Acceleration

Kinematic equations describe the motion of objects undergoing constant acceleration, relating parameters such as displacement, initial and final velocities, acceleration, and time. These equations are useful for determining quantities like final speed or distance traveled during free fall, offering a bridge between the dynamic forces acting on an object and its motion.

Transcript

00:01 So here we can apply newton's second law for part a, and we can say that the gravitational force mg minus the tension t would be equaling the mass times the acceleration.

00:11 And so t would then be equalling the mass multiplied by the difference between the acceleration due to gravity minus the linear acceleration a.

00:22 We know the torque produced tau would be equaling the force, the tension force, multiplied by r.

00:30 This would be r the radius of the drum so t r equaling the moment of inertia multiplied by the angular acceleration by the definition of torque and we can say that then m multiplied by g minus a multiplied by r equaling the moment of inertia i multiplied by the linear acceleration divided by r substituting in for the angular acceleration.

01:02 And so we have then m .g .r squared minus m .a .r squared, equaling i times a.

01:19 And solving for a, this would then be equalling to mgr squared divided by mr squared plus i, the moment of a, for part b, we can say that velocity final squared minus velocity initial squared equaling 2 times a times h the height.

01:53 We know that the initial velocity is going to be zero, and so we can solve for the acceleration a, this would be equaling v squared over 2h.

02:04 We can solve.

02:06 This would be 4 .0 meters per second, quantity squared, divided by 2 multiplied by 20 meters, and this is giving us then 0 .4 meters per second squared.

02:25 We can say that then solving for i, i would then be equaling to mgr squared divided by a minus mr squared.

02:42 And so this would essentially be equaling to mr squared.

02:46 Squared, multiplied by g over a minus 1.

02:54 And so the moment of inertia i would be equaling 50 kilograms, multiplied by 0 .5 meters quantity squared, multiplied by 9 .8 meters per second squared, divided by 0 .4 meters per second squared minus 1.

03:16 And so this is giving us then 294 kilograms meters squared.

03:25 For part c, we can find then the angular acceleration.

03:30 This would be equaling to a over r.

03:33 This is giving us then 0 .4 meters per second squared, divided by 0 .5 meters...

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