The circus apparatus known as the trapeze consists of a bar...

The circus apparatus known as the trapeze consists of a bar suspended by two parallel ropes, each of length $\ell$. The trapeze allows circus performers to swing in a vertical circular arc (Fig. P8.25). Suppose a performer with mass $m$ and holding the bar steps off an elevated platform, starting from rest with the ropes at an angle of $\theta_{i}$ with respect to the vertical. Suppose the size of the performer's body is small compared with the length $\ell$, that she does not pump the trapeze to swing higher, and that air resistance is negligible. (a) Show that when the ropes make an angle of $\theta$ with respect to the vertical, the performer must exert a force $$ F=m g\left(3 \cos \theta-2 \cos \theta_{i}\right) $$ in order to hang on. (b) Determine the angle $\theta_{i}$ at which the force required to hang on at the bottom of the swing is twice the performer's weight.

The circus apparatus known as the trapeze consists of a bar suspended by two parallel ropes, each of length $\ell$. The trapeze allows circus performers to swing in a vertical circular arc (Fig. P8.25). Suppose a performer with mass $m$ and holding the bar steps off an elevated platform, starting from rest with the ropes at an angle of $\theta_{i}$ with respect to the vertical. Suppose the size of the performer's body is small compared with the length $\ell$, that she does not pump the trapeze to swing higher, and that air resistance is negligible. (a) Show that when the ropes make an angle of $\theta$ with respect to the vertical, the performer must exert a force
$$
F=m g\left(3 \cos \theta-2 \cos \theta_{i}\right)
$$
in order to hang on. (b) Determine the angle $\theta_{i}$ at which the force required to hang on at the bottom of the swing is twice the performer's weight.

Key Concepts

Conservation of Energy

This principle is used to relate the gravitational potential energy at different positions with the kinetic energy gained or lost by the system. In problems involving swinging objects or pendulums, the change in height corresponds to a change in potential energy, which directly translates into kinetic energy as the object swings downward. This concept is crucial for determining the speeds at various points along the circular arc.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path. In the case of a swinging performer or pendulum, this force is provided by the tension in the ropes or rod and varies with the speed of the performer. Analyzing the centripetal requirement at different points on the swing is essential in understanding how the forces must be adjusted, such as the needed grip strength at various angles.

Dynamics of Circular Motion

This concept involves applying Newton’s laws to an object undergoing circular motion, where both gravitational and tension forces contribute to the net force acting on the object. Evaluating the balance and components of these forces (especially in the vertical and horizontal directions) allows one to determine the conditions for motion, particularly how the rope’s tension must adjust to provide the necessary centripetal force as the angle of the swing changes.

Force Analysis in Pendulum Motion

This concept focuses on the breakdown of forces acting on a pendulum-like system. By resolving the gravitational force into components along and perpendicular to the direction of the rope, and considering the tension that must counteract both gravitational pull and provide centripetal acceleration, one can derive relationships between the applied forces and the swing angle. This analysis is key to understanding how the force requirements change as the pendulum moves from its initial position to other points along its trajectory.

Transcript

00:02 All right, so for this problem, we're going to start by just thinking about how the trapeze is moving.

00:14 And so it's starting at an angle of the theta 1.

00:17 And so the potential energy it's going to have at this point due to gravity will be due to this height right here, which we can find in terms of the length of the trapeze is l, which is the maximum or the minimum height.

00:34 Minus l cosine theta, which is the projection of this triangle right here.

00:40 As we swing down, this height's going to decrease, and we're going to get a new height, which is going to be minus l cosine theta as this theta approaches zero.

00:54 And so we can write a conservation of energy equation starting from here for any point along this path, where each, the initial kinetic energy is going to be zero.

01:07 Initial potential energy is mgh1, which is this one right here, in terms of theta 1.

01:14 The final kinetic energy is 1⁄2mb squared, and the final potential energy is mgh, where h is in terms of theta.

01:25 Now we can plug in these relations right here and solve for v squared.

01:31 If you solve for b squared first, you get 2g times h1 minus h.

01:37 And then you plug in the l's and minus cosines.

01:42 We can pull the l out, factor that up.

01:45 We get 1 minus cosine theta minus l minus, or 1 minus cosine theta.

01:52 So cosine theta 1 minus cosine theta, which minus the minus the minus becomes positive.

01:57 The ones cancel each other out.

02:00 And we end up with this.

02:03 Is equal to 2gl, cosine theta, minus cosine theta 1.

02:08 From here, we can look at the sum of the forces on the trapeze artist as she swings down...

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