Problem 19 - University Physics with Modern Physics

Question

A 750.0 -kg boulder is raised from a quarry 125 \(\mathrm{m}\) deep by a long uniform chain having a mass of 575 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 8.70 m/s². (b) It takes approximately 5.34 seconds.

Step 1: Calculate the Weight of the Boulder and Chain

First, we need to calculate the total weight of the boulder and the chain. The weight of the boulder is given by\( W_b = m_b \cdot g = 750.0 \times 9.81 \),where \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity.The weight of the chain is similarly given by\( W_c = m_c \cdot g = 575 \times 9.81 \).

Step 2: Determine Maximum Total Tension the Chain Can Support

The chain can support a maximum tension 2.50 times its own weight. Therefore, the maximum tension that the chain can support is:\( T_{max} = 2.50 \times W_c = 2.50 \times (575 \times 9.81) \).

Step 3: Calculate the Maximum Net Force Available for Accelerating the Boulder

The maximum tension must support both the boulder and accelerate it out of the quarry. The net force for acceleration \( F_{net} \) is given by:\( F_{net} = T_{max} - W_b \).

Step 4: Find the Maximum Acceleration of the Boulder

Now, use Newton's second law to find the maximum acceleration \( a_{max} \).\( F_{net} = m_b \cdot a_{max} \),so,\( a_{max} = \frac{F_{net}}{m_b} \).

Step 5: Calculate the Time Taken to Lift the Boulder

Since the boulder starts from rest, the equation for motion is\( d = \frac{1}{2} a_{max} t^2 \),where \( d = 125 \, \text{m} \) is the depth of the quarry. Solving for \( t \), we get:\( t = \sqrt{\frac{2d}{a_{max}}} \).

Key Concepts

Understanding Tension

In mechanics, *tension* refers to the force transmitted through a string, cable, or chain when it is pulled tight by forces acting from opposite ends. It is a key concept in this exercise as the chain raises the boulder from the quarry. - The tension within the chain must counteract the weight of both the boulder and itself.
- The chain can withstand a tension up to 2.50 times its own weight before breaking.
To calculate the tension, you start with the weight of the chain. Since the maximum tension is a multiplier of the chain's weight, it's crucial to know the chain's mass, which is 575 kg in this case. The force of gravity acting on this mass is also a factor, using the formula: \[ T = 2.50 imes (m_c imes g) \] where \( m_c \) is the mass of the chain and \( g = 9.81 \, \text{m/s}^2 \) is the gravitational acceleration.

Decoding Acceleration

*Acceleration* indicates how fast an object can change its velocity. In the context of the boulder being lifted, it's how quickly it speeds up as it ascends. - Acceleration is directly linked to the net force acting upon an object.
- It is determined by rearranging Newton's second law to solve for acceleration.
When lifting the boulder, the chain's tension that goes into accelerating it is: \[ F_{net} = T_{max} - W_b \] where \( W_b \) is the weight of the boulder. The acceleration \( a_{max} \) can then be obtained using: \[ a_{max} = \frac{F_{net}}{m_b} \] This formula showcases that the greater the available net force from the tension, the higher the possible acceleration.

Explaining Newton's Second Law

Newton's second law is fundamental in understanding how objects behave when forces are applied. It is stated as \( F = ma \), where \( F \) is the net force applied to an object, \( m \) is the mass, and \( a \) is the acceleration. - The boulder's acceleration depends on the net force derived from the chain's tension.
- If the tension exceeds the weight of the boulder, the excess force accelerates it upward.
Newton's second law allows us to calculate this acceleration by knowing the net force acting on the boulder: \[ a_{max} = \frac{F_{net}}{m_b} \] This simplicity shows the power of Newton's second law. It connects mass, acceleration, and force, serving as a predictive tool to understand how the boulder reacts to tension.

Calculating Weight and its Role

*Weight* is a key factor in mechanics, representing the force due to gravity acting on a mass. The weight of both the boulder and the chain affects the maximum tension required for lifting. - Weight is calculated by multiplying mass by gravitational acceleration: \( W = m \times g \).
- For both the boulder and the chain, the individual weights are needed to sum the forces that the chain must hold.
In this exercise: - Boulder weight: \( W_b = 750.0 \times 9.81 \approx 7357.5 \, \text{N} \) - Chain weight: \( W_c = 575 \times 9.81 \approx 5641.75 \, \text{N} \) These weights ensure that the total force does not exceed the chain's maximum tension, preventing the chain from breaking and allowing proper movement of the boulder.