An object with a weight of 50.0 N is attached to the free end of a light string wrapped around a

An object with a weight of 50.0 N is attached to the free...

An object with a weight of 50.0 $\mathrm{N}$ is attached to the free end of a light string wrapped around a reel of radius 0.250 $\mathrm{m}$ and mass 3.00 $\mathrm{kg}$ . The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center. The suspended object is released 6.00 $\mathrm{m}$ above the floor. (a) Determine the tension in the string, the acceleration of the object, and the speed with which the object hits the floor. (b) Verify your last answer by using the principle of conservation of energy to find the speed with which the object hits the floor.

Key Concepts

Relationship between Linear and Angular Quantities

In problems involving rotation with translation, the linear acceleration of an object and the angular acceleration of the rotating body are directly related by the radius of rotation. Specifically, the linear acceleration is equal to the radius times the angular acceleration (a = r?). This connection is vital for linking the translational motion of a falling object to the rotational motion of a reel or disk.

Conservation of Energy

The principle of conservation of energy states that the total mechanical energy in an isolated system remains constant if only conservative forces are doing work. In dynamics problems, converting between potential energy and kinetic energy (both translational and rotational) provides an alternative method to analyze the motion of objects, allowing for verification of results obtained from the force analysis.

Newton's Second Law for Translation

This principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). It is fundamental when analyzing the vertical motion of objects under gravity, taking into account forces such as weight and tension from a string, and it allows us to set up equations that relate these forces to acceleration.

Rotational Dynamics and Moment of Inertia

Rotational dynamics extends Newton's laws to objects that rotate, by incorporating the concept of torque and moment of inertia. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For objects such as a solid disk, calculating the moment of inertia is crucial for determining the resulting angular acceleration when a torque is applied by forces such as the tension in a string.

Transcript

00:01 In this exercise, we have an object that has a weight m .g of 50 newtons that is tied to a rope that is wrapped around a disk that has a mass of 3 kilograms and 0 .25 meters.

00:17 And initially, the object is 6 meters above the ground.

00:24 In question a, we want to find the tension on the rope, the acceleration of the...

00:31 The object that's falling and the speed with which the object arrives in the ground.

00:43 And we have to take into account that the object starts its movement at rest.

00:50 So basically here what we have to do is to use the principle of torque in the disk, and we have to draw the force diagram on the falling object.

01:08 So first of all, i'm going to draw the force diagram and then write the force equation for the following object.

01:18 Notice that there is only a gravitational force exerting a certain weight that is equal to m g downward and then the rope that exerts a tension t.

01:31 So the equation for the object is mg minus t is equal to m .a.

01:42 Okay.

01:43 This is the equation for the object.

01:46 And then we have to use the concept of torque for the reel for the disk.

01:55 So notice that the only force that acts on the disk is the tension force.

02:02 The only force that causes torque, actually.

02:05 It's the tension force.

02:07 And the torque is the tension force times the radius of the disk.

02:15 And this is equal to the moment of inertia, i of the disk, times the linear, i'm sorry, the angular acceleration alpha.

02:25 Now, i, the moment of inertia of a disk, is m, r squared over 2.

02:31 And the angular acceleration alpha is equal to the linear acceleration a divided by r.

02:38 And notice that both a's here are the same, so the acceleration of the real is the same acceleration of the following object because they're tied by a rope.

02:49 So basically, the system of equations that we have to solve is mg minus t is equal to ma.

02:55 Then t times r is equal to m r squared over two times a over r.

03:06 So this is equal to m r a divided by two.

03:11 And if we just cancel the r's, we have that t is equal to ma over 2.

03:20 So basically i'm going to use these two equations here.

03:23 And i'm going to sum them up...

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