Problem 24 - College Physics

Question

\(\bullet$$\bullet\) A 750.0 -kg boulder is raised from a quarry 125 \(\mathrm{m}\) deep by a long uniform 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 to be lifted out at maximum acceleration if it started from rest?

Step-by-Step Solution

Answer

(a) Maximum acceleration is 0.827 m/s². (b) Time to lift is approximately 17.39 s.

Step 1: Calculate the Gravitational Force

First, calculate the gravitational force acting on the boulder and the chain. Use the formula for gravitational force: \[ F_{gravity} = m imes g \]where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately \( 9.81 \ \text{m/s}^2 \)).For the boulder: \[ F_{boulder} = 750.0 \ \text{kg} imes 9.81 \ \text{m/s}^2 = 7357.5 \ \text{N} \]For the chain: \[ F_{chain} = 575 \ \text{kg} imes 9.81 \ \text{m/s}^2 = 5640.75 \ \text{N} \]The total gravitational force that must be supported is the sum of these two forces.

Step 2: Calculate the Maximum Tension

The maximum tension the chain can support is given by:\[ T_{max} = 2.50 imes F_{chain} \]Plugging in the values:\[ T_{max} = 2.50 imes 5640.75 \ \text{N} = 14101.875 \ \text{N} \]

Step 3: Determine Maximum Acceleration for the Boulder

To find the maximum acceleration, we apply Newton's second law:\[ F_{net} = T_{max} - (F_{boulder} + F_{chain}) = (m_{boulder} + m_{chain}) imes a_{max} \]Substitute the known values:\[ a_{max} = \frac{T_{max} - (F_{boulder} + F_{chain})}{m_{boulder} + m_{chain}} \]\[ a_{max} = \frac{14101.875 \ \text{N} - 13005.25 \ \text{N}}{750.0 \ \text{kg} + 575 \ \text{kg}} = \frac{1096.625}{1325} \ \text{m/s}^2 = 0.827\ \text{m/s}^2 \]

Step 4: Calculate the Time to Raise the Boulder

Use the kinematic equation to find time \( t \) when starting from rest:\[ s = \frac{1}{2} a t^2 \]Solving for \( t \), where \( s = 125 \ \text{m} \) and \( a = 0.827 \ \text{m/s}^2 \):\[ 125 = \frac{1}{2} \times 0.827 \times t^2 \]\[ t^2 = \frac{250}{0.827} \approx 302.18 \]\[ t = \sqrt{302.18} \approx 17.39 \ \text{s} \]

Key Concepts

Newton's Second Law

Newton's Second Law is one of the foundational principles of physics that enables us to understand how objects behave when subjected to various forces. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it is expressed as:\[ F_{net} = m \cdot a \]where:

• \( F_{net} \) is the net force applied on the object.
• \( m \) is the mass of the object.
• \( a \) is the acceleration of the object.

In this exercise, Newton's Second Law is used to determine the maximum possible acceleration of the boulder being lifted from the quarry. By calculating the net force on the boulder – which is the maximum tension minus the gravitational force – and dividing by the combined mass of the boulder and chain, we find the acceleration. This calculation ensures the boulder can be raised as fast as possible without exceeding the chain's maximum tension.

Gravitational Force

Gravitational force is the force of attraction between two masses. On Earth, this force is typically one of the key forces affecting objects. The standard formula used to calculate gravitational force on an object due to Earth's gravity is:\[ F_{gravity} = m \cdot g \]Here, \( g \) represents the acceleration due to gravity, typically about \( 9.81 \, \text{m/s}^2 \) on the Earth's surface. This force acts downwards, pulling objects towards the center of the Earth.In this problem, we find the gravitational forces of both the boulder and the chain.

• The boulder's force: \( F_{boulder} = 750.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 7357.5 \, \text{N} \)
• The chain's force: \( F_{chain} = 575 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 5640.75 \, \text{N} \)

Combining these gives the total gravitational force that needs to be overcome to lift these objects.

Kinematic Equations

Kinematic equations describe the motion of objects without considering the causes of this motion. They are especially useful in situations where an object is accelerated over time. One of the most common kinematic equations used is:\[ s = \frac{1}{2} a t^2 \]where:

• \( s \) is the displacement or distance traveled.
• \( a \) is the acceleration.
• \( t \) is the time taken.

In the exercise, this equation is applied to calculate how long it takes for the boulder to be lifted from rest over a distance of 125 meters with the maximum acceleration calculated. Knowing the maximum acceleration, we can substitute the values and solve for \( t \) to find the lifting time necessary to clear the quarry.

Maximum Tension

The concept of maximum tension is crucial when analyzing forces in mechanical systems. It defines the greatest force a material can withstand without failure. For the chain in this problem, the maximum tension it can endure is 2.5 times its own weight. Given the mass of the chain, we calculate the maximum tensile force it can support:\[ T_{max} = 2.50 \times F_{chain} = 14101.875 \, \text{N} \]This maximum tension ensures the chain remains intact while lifting the boulder. Understanding and calculating max tension allows engineers to ensure that lifting materials and structures operate safely within their limits. In this problem, ensuring the tension does not go beyond this threshold while lifting allows for the maximum safe acceleration of the boulder.