Problem 21 - College Physics

Question

A \(750.0 \mathrm{~kg}\) boulder is raised from a quarry \(125 \mathrm{~m}\) deep by a long chain having a mass of \(575 \mathrm{~kg} .\) This chain is of uniform strength, but at any point it can support a maximum tension no greater than 2.50 times its weight without breaking. (a) What is the maximum acceleration the boulder can have and still get out of the quarry, and (b) how long does it take for the boulder to be lifted out at maximum acceleration if it started from rest?

Step-by-Step Solution

Answer

(a) The maximum acceleration of the boulder is 0.833 m/s². (b) It takes approximately 17.34 seconds to lift the boulder out at maximum acceleration.

Step 1: Calculate the weight of the boulder and the chain

The weight of the boulder is given by the equation \( W_b = m_b \cdot g \), where \( m_b \) is the mass of the boulder, and \( g \) is the acceleration due to gravity \( (9.81 \text{ m/s}^2) \). Thus, the weight of the boulder \( W_b = 750.0 \text{ kg} \times 9.81 \text{ m/s}^2 = 7357.5 \text{ N} \). Similarly, the weight of the chain \( W_c = 575.0 \text{ kg} \times 9.81 \text{ m/s}^2 = 5640.75 \text{ N} \).

Step 2: Determine the maximum tension the chain can support

The chain can support a tension that is 2.50 times its weight without breaking. Thus, \( T_{max} = 2.50 \times W_c = 2.50 \times 5640.75 \text{ N} = 14101.875 \text{ N} \).

Step 3: Set up the equation for maximum tension

The total force acting on the system, considering the maximum acceleration \( a \), is given by \( T_{max} = W_b + W_c + (m_b + m_c) \cdot a \). Rearranging gives the equation for \( a: a = \frac{T_{max} - (W_b + W_c)}{m_b + m_c} \). Where \( m_c = 575 \text{ kg} \).

Step 4: Calculate the maximum acceleration

Substitute the known values into the equation: \( a = \frac{14101.875 \text{ N} - (7357.5 \text{ N} + 5640.75 \text{ N})}{750.0 \text{ kg} + 575.0 \text{ kg}} = \frac{14101.875 \text{ N} - 12998.25 \text{ N}}{1325.0 \text{ kg}} = 0.833 \text{ m/s}^2 \).

Step 5: Calculate the time required to lift the boulder

Using the kinematic equation for constant acceleration \( d = \frac{1}{2} a t^2 \) and initial velocity zero, substitute \( d = 125 \text{ m} \) and \( a = 0.833 \text{ m/s}^2 \): \(125 = \frac{1}{2} \times 0.833 \times t^2 \), \( t^2 = \frac{125 \times 2}{0.833} \), \( t = \sqrt{\frac{250}{0.833}} \approx 17.34 \text{ s} \).

Key Concepts

Classical Mechanics

In classical mechanics, we explore the forces that govern the motion of objects in our everyday world. This branch of physics, developed by scientists like Isaac Newton, helps us understand how objects move and interact using laws and equations. One of the key contributions of classical mechanics is Newton’s Laws of Motion, which explain how forces affect motion and equilibrium.

### Newton’s Laws of Motion
- **First Law (Law of Inertia):** An object in motion stays in motion unless acted upon by a net external force. Similarly, an object at rest stays at rest.
- **Second Law (Law of Acceleration):** The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, given by the equation: \( F = m imes a \).
- **Third Law (Action and Reaction):** For every action force, there is an equal and opposite reaction force.

These principles together provide a framework to solve problems involving motion, such as the lifting of a boulder out of a quarry. Understanding these laws allows us to calculate the required forces and motion parameters like tension and acceleration.

Kinematics

Kinematics is the study of motion without considering the forces that cause it. It focuses on the different aspects of motion such as speed, velocity, and acceleration. In the problem above, kinematics equations help us determine how an object moves under certain conditions.

### Key Kinematic Equations
- **Displacement (d):** The change in position of an object. It can be calculated for constant acceleration using the equation \( d = v_i imes t + \frac{1}{2} a imes t^2 \), where \( v_i \) is the initial velocity, \( t \) is time, and \( a \) is acceleration.
- **Velocity and Acceleration:** Velocity is the speed of an object in a given direction. Acceleration is the change of velocity over time.

For the boulder in our problem, we started with rest, which implies initial velocity, \( v_i \), is zero. We then applied the kinematic equation to find the time it takes for the boulder to be lifted with a constant maximum acceleration.

Forces and Tension

Forces are any interactions that cause an object to change its motion. Tension is a specific type of force that is transmitted through a string, cable, or chain when it is pulled tight by forces from opposite ends. In our scenario, both the weight of the chain and the boulder contribute to the total force that must be overcome for motion to occur.

### Understanding Tension
- **Weight as a Force:** The weight of an object is a force calculated by \( W = m imes g \), where \( g \) is the acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\) on Earth).
- **Maximum Tension:** The chain can only hold a force up to a certain limit before it breaks. Here, it can withstand a tension up to 2.50 times its own weight.

In practice, solving for force and tension requires using their interrelatedness. The weight of the boulder and chain, coupled with the tension limit of the chain, provides the necessary conditions to calculate maximum acceleration.

Acceleration and Motion

Acceleration is the rate at which an object's velocity changes. Motion involves the activities or paths objects take when subject to forces and acceleration. In physics, understanding acceleration is crucial for predicting how and when an object will move.

### Calculating Acceleration
- **Derived Formula:** To find the maximum acceleration that doesn't exceed the tension capability of the chain, we rearrange the standard force equation to \( a = \frac{T_{max} - (W_b + W_c)}{m_b + m_c} \). Here, \( T_{max} \) is the maximum tension the chain can endure.
- **Constant Acceleration:** With a known force, objects reveal predictable paths and times required for travel. This aligns with the kinematic equation for objects at rest: \( d = \frac{1}{2} a t^2 \).

In this exercise, calculating the maximum acceleration ensures that the boulder can be safely lifted out of the quarry without risking the chain's integrity. This real-world application of acceleration examines practical thresholds to ensure efficient and safe motion.