Problem 66 - University Physics with Modern Physics

Question

You complain about fire safety to the landlord of your high-rise apartment building. He is willing to install an evacuation device if it is cheap and reliable, and he asks you to design it. Your proposal is to mount a large wheel (radius 0.400 m) on an axle at its center and wrap a long, light rope around the wheel, with the free end of the rope hanging just past the edge of the roof. Residents would evacuate to the roof and, one at a time, grasp the free end of the rope, step off the roof, and be lowered to the ground below. (Ignore friction at the axle.) You want a 90.0-kg person to descend with an acceleration of g/4. (a) If the wheel can be treated as a uniform disk, what mass must it have? (b) As the person descends, what is the tension in the rope?

Step-by-Step Solution

Answer

(a) The wheel must have a mass of 120 kg. (b) The tension in the rope is approximately 661.5 N.

Step 1: Understand the Problem

We need to calculate the mass of the wheel required to ensure a 90.0 kg person descends with an acceleration of \( \frac{g}{4} \). We'll model the wheel as a uniform disk, which means its moment of inertia \( I \) is given by \( \frac{1}{2} m R^2 \), where \( m \) is the mass of the wheel and \( R \) is the radius (0.400 m). Next, we find the tension in the rope during the person's descent.

Step 2: Identify Forces and Equations

Consider the forces acting on the person and the wheel. The person's weight is \( mg \), and the tension \( T \) in the rope acts upward. Newton's second law gives \( mg - T = ma \), where \( a = \frac{g}{4} \). For the wheel, the torque \( \tau = TR \) results in an angular acceleration \( \alpha = \frac{a}{R} \), with \( \tau = I\alpha \).

Step 3: Relate Linear and Rotational Motion

Using \( TR = I \frac{a}{R} \), substitute \( I \) from the disk formula: \( TR = \left( \frac{1}{2} m R^2 \right) \frac{a}{R} \). Simplify to find \( T = \frac{1}{2} ma \).

Step 4: Solve for Wheel Mass 'm'

With \( mg - T = ma \): replace \( T \) with \( \frac{1}{2} ma \), to get \( mg - \frac{1}{2} ma = ma \). Simplifying, \( m = \frac{4M}{3} \), where \( M = 90 \) kg is the person's mass. Calculate \( m = \frac{4 \times 90}{3} = 120 \) kg.

Step 5: Calculate Tension in the Rope

Using \( T = mg - ma \), substitute for \( a = \frac{g}{4} \): \( T = 90g - 90 \frac{g}{4} \). Simplify the expression: \( T = 90g - 22.5g = 67.5g \). Compute \( T = 67.5 \times 9.8 \approx 661.5 \) N.

Key Concepts

Mechanics

Mechanics is a branch of physics that deals with the motion of objects and the forces that affect that motion. In this exercise, we are designing an evacuation device using principles from mechanics to ensure a safe descent for individuals.

Key aspects of mechanics relevant to this problem include:

• Mass and Weight: Mass is a measure of the amount of matter in an object and does not change depending on location. Weight is the force exerted by gravity on that mass and is dependent on the local gravitational field.
• Acceleration: This is the rate at which an object's velocity changes. Here, we calculate the required acceleration for a person descending with a controlled rate.
• Tension: This is the force exerted by the rope on the person. Since the person is descending, we must calculate tension in the rope to ensure it is within safe limits and helps control the descent.

By understanding these mechanics principles, we can ensure that the wheel's mass is appropriately set and the tension is controlled, big parts of ensuring a smooth and safe evacuation process.

Rotational Motion

Rotational motion involves objects rotating around a central point or axis. In our evacuation device, the wheel undergoes rotational motion.

Important concepts in rotational motion include:

• Radius: The distance from the center of the wheel to its edge, which in this problem is 0.400 meters. This radius affects how the wheel turns and is critical for calculating other rotational properties.
• Moment of Inertia: This is analogous to mass in linear motion and measures the resistance to changes in rotational motion. For a disk, the moment of inertia is calculated using the formula \( I = \frac{1}{2} m R^2 \), where \( m \) is the mass of the wheel.
• Torque and Angular Acceleration: Torque is the twisting force that causes rotational acceleration. It is determined by the tension in the rope and the wheel's radius, expressed as \( \tau = TR \). The relationship between linear acceleration \( a \) and angular acceleration \( \alpha \) is \( \alpha = \frac{a}{R} \).

By understanding the interplay of these rotational motion concepts, we design the wheel to provide a steady descent, ensuring the safety of the people evacuating.

Newton's Second Law

Newton's Second Law of motion states that the force acting on an object is equal to the mass of that object multiplied by its acceleration \( F = ma \). This principle is central to solving our evacuation device problem.

Key points in utilizing Newton's Second Law include:

• Differentiating Overall Forces: We calculate forces acting on the person (gravitational force), and on the wheel (rotational force), balancing them to find the wheel's required mass, ensuring orderly descent.
• Calculating Net Force: For the person lowering down, net force is the difference between the gravitational pull and rope tension: \( mg - T = ma \). By rearranging, we solve for tension and verify the descent.
• Coupling Linear and Rotational Components: Linking the linear equations of motion (descent of the person) with rotational dynamics (turning of the wheel) via Newton’s second law, provides insights on required mass for limiting acceleration.

Applying Newton's Second Law ensures proper force balance between person and wheel, dictating safe and controlled operation of our proposed evacuation device.