Problem 31 - University Physics with Modern Physics

Question

A 5.60-kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 N. If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 12.0 m without breaking the cord?

Step-by-Step Solution

Answer

The minimum time required is approximately 2.58 seconds.

Step 1: Identify the forces involved

Determine the forces acting on the bucket: the gravitational force (weight) and the tension force in the cord. The gravitational force is given by the formula: \( F_g = m \cdot g \), where \( m = 5.60 \text{ kg} \) is the mass of the bucket, and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity.

Step 2: Calculate the gravitational force

Using the formula \( F_g = m \cdot g \), we find the gravitational force: \( F_g = 5.60 \times 9.8 = 54.88 \text{ N} \).

Step 3: Determine maximum tension for acceleration

Since the breaking strength of the cord is 75.0 N, the maximum net force for safe acceleration is \( 75.0 \text{ N} - 54.88 \text{ N} = 20.12 \text{ N} \). This is the maximum upward force before the cord breaks.

Step 4: Apply Newton's Second Law

Use \( F = m \cdot a \) to find the maximum acceleration the bucket can have without breaking the cord: \( 20.12 = 5.60 \cdot a \). Solving for \( a \), we get \( a = \frac{20.12}{5.60} = 3.59 \text{ m/s}^2 \).

Step 5: Calculate the minimum time for the vertical motion

Use the kinematic equation for motion: \( d = \frac{1}{2} a t^2 \), where \( d = 12.0 \text{ m} \). Substituting the values, we get \( 12.0 = \frac{1}{2} \times 3.59 \times t^2 \). Simplify to find \( t^2 = \frac{12.0}{1.795} \) which results in \( t^2 = 6.68 \). Find \( t \): \( t = \sqrt{6.68} \approx 2.58 \text{ seconds} \).

Key Concepts

Gravitational Force

Gravitational force is a fundamental force of nature that acts between two masses. This is the force that makes apples fall from trees and keeps planets in orbit around the sun. In mechanics problems, especially those involving objects in motion, understanding gravitational force is crucial. Gravitational force on an object near the Earth's surface is calculated using the equation: \[ F_g = m \cdot g \] where:

• \( F_g \) is the gravitational force or weight.
• \( m \) is the mass of the object, measured in kilograms (kg).
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \) on Earth.

For the 5.60 kg bucket example, the gravitational force is calculated to be \( 54.88 \, \text{N} \). This means the Earth pulls the bucket downward with a force of 54.88 N. Understanding this allows you to analyze and solve how other forces, such as tension, interact with this gravitational pull.

Kinematics Equations

Kinematics equations are used to describe the motion of objects. They allow us to calculate unknown variables, such as time, distance, acceleration, and velocity, when certain conditions are known. When dealing with vertical motion and constant acceleration in mechanics problems, the relevant kinematic equation is: \[ d = \frac{1}{2} a t^2 \] where:

• \( d \) is the distance traveled, measured in meters (m).
• \( a \) is acceleration, in meters per second squared (\( \text{m/s}^2 \)).
• \( t \) is time, in seconds (s).

In the bucket exercise, we resolved the maximum acceleration, \( a = 3.59 \, \text{m/s}^2 \), and the initial position, speed (starting from rest), and distance \( d = 12.0 \, \text{m} \) were considered to determine the minimum time \( t \) to safely raise the bucket. Plugging these into the equation helps find \( t = 2.58 \, \text{s} \). These concepts are widely applied, from basketball shots to vehicles speeding up.

Mechanics Problems

Mechanics problems involve understanding how forces affect the motion of objects, using principles laid down by Sir Isaac Newton. Solving these problems typically requires:

• Identifying all forces acting on an object, in this case, gravitational force and tension.
• Applying Newton's Second Law, \( F = m \cdot a \), to relate forces to acceleration.
• Using kinematics to connect acceleration, distance, and time.

A common challenge in these problems is determining how much force can be safely applied without overstepping limits, such as breaking the cord in the exercise. Here, careful calculations ensured that the force did not exceed the 75 N breaking strength, using realistic physics rules. Solving mechanics problems trains skills like logical thinking and quantitative reasoning, valuable for fields like engineering and architecture.