Rotational Equilibrium and Rotational Dynamics

[ataglance05]

Homework Statement

Question Details:
A 10 kg engine is rotating at the rate of 20 rad/sec about a point on a wire 3 meters in length with a working tensile strength of 1.8*106 N. The engine is fired and produces an acceleration of 1 rad/sec2. What's the maximum possible ω (angular velocity) (in other words: 1) whats's the fastest ω which doesn't break the wire?) 2)How long should the engine be fired to reach the maximum possible ω? 3)How far will the engine travel in meters while it's accelerating to the maximum possible ω?

Homework Equations

The Attempt at a Solution

I believe it's right, however this is only #1. Need help on #2 and #3.

Thanks!

 

[andrevdh]

2 can be solved with the equation

$$w_f = w_i + \alpha t$$

since the angular acceleration is sustained as long as the engine is firing. The intial angular speed is 20 and the final is 245 rad/s. Once you solved for the time use it in the other equation that will give the angle in radians that it turned through during the burn. Using the radius you can then calculate the total distance covered during this time.

 

[m2003]

I would like to provide a response to the content presented in this question. The topic of rotational equilibrium and rotational dynamics is a fundamental concept in physics that deals with the motion of objects rotating around an axis or point. In this problem, we are given a 10 kg engine rotating at a rate of 20 rad/sec about a point on a wire with a length of 3 meters and a tensile strength of 1.8*10^6 N. The engine is then fired, causing an acceleration of 1 rad/sec^2. We are asked to find the maximum possible angular velocity (ω) that the engine can reach without breaking the wire, the time it takes for the engine to reach this maximum ω, and the distance the engine will travel while accelerating to this maximum ω.

To solve this problem, we can use the principles of rotational equilibrium and dynamics. The first step is to determine the tension in the wire, which can be found using the formula T = mgcosθ, where T is the tension, m is the mass of the engine, g is the acceleration due to gravity, and θ is the angle between the wire and the vertical axis. In this case, θ is equal to 90 degrees, so the tension in the wire is simply equal to the weight of the engine, which is 10 kg * 9.8 m/s^2 = 98 N.

Next, we can use the equation τ = Iα to find the torque (τ) acting on the engine, where I is the moment of inertia and α is the angular acceleration. The moment of inertia for a point mass rotating around a fixed axis is equal to mr^2, where m is the mass and r is the distance from the axis of rotation. In this case, the moment of inertia is equal to 10 kg * (3 m)^2 = 90 kgm^2. Substituting these values into the equation, we get τ = 90 kgm^2 * 1 rad/sec^2 = 90 Nm.

Now, we can use the equation τ = Fr to find the force (F) acting on the engine, where r is the radius of rotation. In this case, the radius is equal to 3 meters, so F = 90 Nm / 3 m = 30 N. This force is provided by the tension in the wire, so we can set up