Problem 13.61 Part A At the instant θ = 45^∘, the boy's...

Problem 13.61 Part A At the instant $\theta = 45^\circ$, the boy's center of mass G is momentarily at rest. (Figure 1) Determine his speed when $\theta = 90^\circ$. The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords. Express your answer to three significant figures and include the appropriate units. $v = 12.27 \frac{ft}{s}$

Transcript

00:01 In this exercise, we're going to be talking about circular motion.

00:03 So before actually solving the problem, i'm going to briefly review the topic.

00:08 So when we're talking about a circular motion, we have a particle that undergoes a circular trajectory.

00:15 And the acceleration of this particle can be decomposed into two components.

00:20 The first one is radial, meaning that it points towards the center of the circle of the trajectory.

00:30 And this is called the centripetal acceleration.

00:34 And we also have the tendential acceleration that is tangential to the curve.

00:43 Okay.

00:44 And we have that the centripetal acceleration is equal to v squared over r.

00:49 V is the speed of the particle while r is the radius of the orbit, while the tangential acceleration is equal to the derivative of the speed.

01:05 With respect to time.

01:07 So the centripetal acceleration is all about changes in the direction of the velocity while even if the velocity, even if the speed remains the same, while the tangential acceleration is all about the changes in the speed, that is the changes in the magnitude of the velocity.

01:32 So this is the information we need in order to solve the exercise.

01:36 So what we have is a boy that is in a swing.

01:42 The boy has a mass of 60 pounds and initially makes an angle theta with a horizontal of 60 degrees as shown in the picture.

01:52 The length of the wire that connects the swing to the center of rotation is 10 feet.

02:02 And the initial speed of the boy is zero meters per second.

02:06 So instantaneously at theta 0 equals 60 degrees, the speed is 0.

02:14 And our goal is to use this information in order to find what is the speed of the boy when theta is equal to 90 degrees.

02:25 Okay.

02:27 And also, we need to find what is the tension on each cable on the swing at the same instant.

02:38 So remember that the swing is composed of two.

02:40 Cables, each one with the tension t.

02:44 Okay, so in order to solve this, we need to draw a force diagram on the boy.

02:50 So here we have the boy represented by this black circle.

02:57 We have the tension force acting in the direction of the swing.

03:07 And the force is equal to two times the individual tension of each cable.

03:15 We also have the gravitational force acting downwards.

03:21 And this is all we have.

03:24 Now notice that we can decompose this force, decompose the forces into two components, one that is centripetal, that is the points in the direction of the center of rotation, and the other one that is radial.

03:46 Okay...

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