Figure P6.57 shows a photo of a swing ride at an amusement...

Figure P6.57 shows a photo of a swing ride at an amusement park. The structure consists of a horizontal, rotating, circular platform of diameter D from which seats of mass $m$ are sus- pended at the end of massless chains of length d. When the system rotates at constant speed, the chains swing outward and make an angle $\theta$ with the vertical. Consider such a ride with the following parameters: $D=$ $8.00 \mathrm{m}, d=2.50 \mathrm{m}, m=$$10.0 \mathrm{kg},$ and $\theta=28.0^{\circ} .$ (a) What is the speed of each seat? (b) Draw a diagram of forces acting on a $40.0-\mathrm{kg}$ child riding in a seat and $(\mathrm{c})$ find the tension in the chain.

Figure P6.57 shows a photo of a swing ride at an amusement park. The structure consists of a horizontal, rotating, circular platform of diameter D from which seats of mass $m$ are sus-
pended at the end of massless chains of length d. When the system rotates at constant speed, the chains swing outward and make an angle $\theta$ with the vertical. Consider such a ride with the following parameters: $D=$ $8.00 \mathrm{m}, d=2.50 \mathrm{m}, m=$$10.0 \mathrm{kg},$ and $\theta=28.0^{\circ} .$ (a) What is the speed of each seat? (b) Draw a diagram of forces acting on a $40.0-\mathrm{kg}$ child riding in a seat and $(\mathrm{c})$ find the tension in the chain.

Key Concepts

Force Decomposition

Force decomposition involves breaking a force into its component parts along chosen axes, typically vertical and horizontal. This technique is essential when dealing with inclined forces or situations where forces act at angles, as it allows the application of equilibrium conditions in each direction separately. In the context of a swinging ride, the tension in the chain is resolved into vertical and horizontal components, balancing the gravitational force and providing the necessary centripetal force.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path. It acts radially inward towards the center of the circle. In rotational problems, it is typically calculated using the formula that relates mass, speed, and radius of the circular path. Recognizing the need for a centripetal force and using it as the balance of forces in the horizontal direction is crucial in analyzing the dynamics of rotating systems such as amusement park rides.

Uniform Circular Motion

This concept describes the motion of an object traveling in a circular path at a constant speed, where the direction of the velocity changes continuously. The analysis involves understanding that even with constant speed, there is an acceleration directed towards the center of the circle, known as centripetal acceleration. This acceleration is provided by a net inward force, which is essential in solving problems involving rotating systems.

Free-Body Diagram

A free-body diagram is a graphical tool used to visualize the forces acting on an object. It involves isolating the object and representing all the forces with arrows, indicating their magnitudes and directions. This technique is vital for understanding how different forces, such as gravitational force and tension, interact in systems undergoing motion, and is particularly useful when applying Newton’s second law of motion.

Transcript

00:01 Problem we are considering a swing ride and what this scenario consists of is first there is a rotating platform with radius four meters this rotating platform right here for the edges of the rotating platform we have some cables of length d and at the end of the cables we had the seats with mass m so we're given the the radius of four meters the mass of the six ten kilograms and the length of the cable is 2 .5 meters so and then when the ride starts spinning the cables start making an angle with the vertical in the way shown it's strong here.

00:36 So we're giving the degree we've given theta is 28 degrees.

00:39 So what important is consider is that the circular motion of the mass does not have a radius of r.

00:46 The actual radius of the sitz circular trajectory is given by r plus d sine theta.

00:52 So this distance here plus d sine theta this distance here.

00:57 So we want to find the speed of the sid.

01:00 In order to do that, we just need to consider, okay, let's do the forces on the map on the horizontal axis and the vertical axis.

01:08 So on the horizontal axis, you just have this equation right here, where p is just the tension on the wire.

01:15 Give it like this, it's just a centripetal force.

01:19 On the vertical axis, we just have this equation.

01:21 It's just the weight and the other component of the tension of the cable.

01:25 So to find the speed we're going to do, we're going to take these two equations and divide them by each other...

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