Question

Figure \(9-54\) shows the massive shield door at a neutron test facility at Lawrence Livermore Laboratory; this is the world's heaviest hinged door. The door has a mass of \(44,000 \mathrm{~kg}\), a rotational inertia about its hinge line of \(8.7 \times 10^{4} \mathrm{~kg} \cdot \mathrm{m}^{2}\), and a width of \(2.4 \mathrm{~m}\). What steady force, applied at its outer edge at right angles to the door, can move it from rest through an angle of \(90^{\circ}\) in \(30 \mathrm{~s}\) ?

Short answer

The steady force required, applied at its outer edge at right angles to the door, can be calculated using these steps. Due to variations depending on the exact values calculated for angular acceleration and torque, it's recommended to replicate the calculations for the exact force needed.

Step-by-step solution

Find angular displacement in radians

First, convert angular displacement from degrees to radians. Use the conversion factor \( \frac{\pi}{180} \) since one full circle equals \(2\pi\) radians. Angular displacement (\( \theta \)) of \(90^{\circ}\) equals \( \frac{\pi}{2} \) radians.

Calculate angular acceleration

Use the equation of motion in rotational form to find angular acceleration. The equation is \( \theta = \omega_i*t + \frac{1}{2}*\alpha*t^{2} \), where \( \theta \) is angular displacement, \( \omega_i \) is initial angular speed, \( \alpha \) is angular acceleration, and \( t \) is time. Given that the door started from rest, initial angular speed is zero. Therefore, rearrange the equation to \( \alpha = \frac{2*\theta}{t^{2}} \). Substitute \( \theta = \frac{\pi}{2} \) rad and \( t = 30 \) s and solve for \( \alpha \).

Calculate torque

Then, use the rotational form of Newton's second law to find the net torque. The equation is \( \tau = I*\alpha \), where \( \tau \) is the net torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration. Substitute \( I = 8.7*10^{4} \) kg*m^2 and the value of \( \alpha \) from the previous step.

Determine force

Finally, use the definition of torque to find the force required. The equation is \( \tau = F*r \) where \( F \) is the force, and \( r \) is the lever arm. In this case, force is applied at the outer edge of the door, so lever arm equals to the width of the door (2.4 m). Rearrange to \( F = \frac{\tau}{r} \), substitute values of \( \tau \) and \( r \), and solve for \( F \).

Key concepts

Angular Displacement

Angular displacement is the angle through which a point or line has been rotated in a specified sense about a specified axis. For example, when a shield door at the neutron test facility moves, we consider its swing from the initial position to its current position as the angular displacement. To get the angular displacement of the door described in our exercise problem, we convert the given 90-degree turn into radians because radians are the standard unit of angular measurement in physics. This conversion is essential for utilizing equations of rotational motion.

Angular Acceleration

In rotational motion physics, just as objects moving in a straight line can accelerate, so too can objects rotating. Angular acceleration is the rate at which the angular velocity of an object changes with respect to time. It's denoted by the Greek letter alpha (\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\(\r\text{\ralpha\r\text{\r\text{}\r\text{\r\)}}}}}}}}}}}}}})) and is measured in radians per second squared (\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r\text{\r{\frac{\r\text{\rradians\r\text{\r\text{}}}}{\r\text{\rradians\r\text{\rr^2\r\text{\r\text{}}}}}}}}}}}}}}))). Applying the provided equation of motion customized for rotational movement to the exercise, we can calculate the angular acceleration required for the massive door to reach a certain angular displacement in the specified time. It's crucial because it helps us determine the necessary force to apply to achieve the desired rotation without overshooting or falling short of the goal.

Torque

Torque is a measure of how much a force acting on an object causes that object to rotate. Imagine it as the rotational equivalent of force. It depends not only on the force applied but also on the distance from the pivot point (or hinge) to the point where the force is applied. This is known as the lever arm. By using the door's moment of inertia and the calculated angular acceleration, you obtain the net torque necessary for its rotation. This concept is directly tied to the effectiveness of the force applied and where it is applied on the door to create the desired angular acceleration.

Moment of Inertia

The moment of inertia, symbolized as 'I', effectively represents rotational mass. If we consider Newton's second law—mass resists linear acceleration—we can analogously state that the moment of inertia resists angular acceleration. In the context of our exercise, the door's moment of inertia is substantial, given its weight and size, and directly influences how much torque is needed to accelerate it to a certain angular velocity. It can be considered the rotational equivalent of mass in the discussion of Newton's second law, where torque is analogous to force, and angular acceleration replaces linear acceleration.