Problem 41 - University Physics with Modern Physics

Question

A \(4.80-\mathrm{kg}\) bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 \(\mathrm{N}\) . (a) Draw the free-body force diagram for the bucket. In terms of the forces on your diagram, what is the net force on the bucket? (b) Apply Newton's second law to the bucket and find the maximum upward acceleration that can be given to the bucket without breaking the cord.

Step-by-Step Solution

Answer

The maximum upward acceleration is approximately 5.81 m/s².

Step 1: Identify Forces

To solve the problem, start by identifying and drawing the forces acting on the bucket. There are two forces: the gravitational force (weight) acting downwards and the tension in the cord acting upwards. Represent these forces in a free-body diagram: gravity (\(F_g = m \cdot g \), where \( m = 4.80 \,\mathrm{kg} \) and \( g = 9.81 \,\mathrm{m/s^2} \)) and tension (\( T \)).

Step 2: Calculate Gravitational Force

The gravitational force can be calculated using the formula \( F_g = m \cdot g \). Substitute the values to get:\[F_g = 4.80 \,\mathrm{kg} \times 9.81 \,\mathrm{m/s^2} = 47.1 \,\mathrm{N}\]

Step 3: Express Net Force

The net force acting on the bucket in terms of the forces in the diagram is given by the difference between the upward tension and the downward gravitational force: \( F_{net} = T - F_g \). This is because tension tries to lift the bucket while gravity pulls it down.

Step 4: Apply Newton's Second Law

Newton's second law states \( F_{net} = m \cdot a \), where \( a \) is the acceleration. Using the expression for the net force from Step 3, we have:\[ T - F_g = m \cdot a \]

Step 5: Solve for Maximum Acceleration

Substitute the maximum tension the cord can withstand (\( T = 75.0 \,\mathrm{N} \)) and the gravitational force into the equation from Step 4:\[ 75.0 \,\mathrm{N} - 47.1 \,\mathrm{N} = 4.80 \,\mathrm{kg} \cdot a \]\[ 27.9 \,\mathrm{N} = 4.80 \,\mathrm{kg} \cdot a \]Solve for \( a \):\[ a = \frac{27.9 \,\mathrm{N}}{4.80 \,\mathrm{kg}} \approx 5.81 \,\mathrm{m/s^2} \]

Step 6: Conclusion

The maximum acceleration without breaking the cord is \( 5.81 \,\mathrm{m/s^2} \).

Key Concepts

Free-Body Diagram

A free-body diagram is a simple visual tool to help us understand the forces acting on an object. By clearly illustrating all the forces at play, it helps us apply the principles of Newton's laws of motion.

To draw a free-body diagram, start with a dot or a simple shape like a circle to represent the object. For the bucket of water weighing 4.80 kg, you'll include two main forces:

• Gravitational Force: This force points downwards, indicating gravity's pull. It's calculated as the product of the mass of the object and the acceleration due to gravity \( (F_g = m \cdot g) \).
• Tension in the Cord: This force is directed upwards. It represents the pull exerted by the cord to lift the bucket.

By labeling these forces accurately on the diagram—ensuring arrows are scaled proportionally to each force's magnitude—it becomes easier to visually analyze the forces and solve problems like those outlined in Newton's Second Law. Such diagrams do not just show strengths but also the directions of forces, setting a baseline for calculating net forces and figuring out the resultant influence on the object's motion.

Gravitational Force

The gravitational force is an ever-present force acting on all objects with mass. It's what gives objects weight and causes them to fall towards the Earth when dropped. For any object, such as our bucket of water, this force can be computed using the formula \( F_g = m \cdot g \), where:

• \( m \) is the mass of the object, in kilograms (kg).
• \( g \) stands for the acceleration due to gravity, approximately \( 9.81 \,\mathrm{m/s^2} \) on Earth's surface. This value can vary slightly based on location.

In our case, with a 4.80 kg bucket, the gravitational force results in a force pulling downward with a magnitude of \( 47.1 \,\mathrm{N} \) (\text{Newtons}).

This force plays a vital role in the problem as it influences both the net force and the acceleration we aim to calculate. Understanding gravitational force is key to analyzing how objects will move or stay in place under various conditions. It is a fundamental force that determines how much support or tension is necessary when lifting or holding an object still.

Tension in a Cord

Tension arises when a cord, rope, or string is used to pull or support an object. It's a force that acts to counterbalance other forces, like gravity, enabling us to lift objects or keep them suspended. In the scenario of our water bucket, tension in the cord must overcome the gravitational force to accelerate the bucket upward.

For this exercise, the tension is noted to have a maximum breaking strength of 75.0 \( \mathrm{N} \). It needs to be managed correctly to keep the bucket lifted without snapping the cord.

According to Newton's Second Law, the net force \( F_{\text{net}} \) can be described by \( T - F_g \) when tension lifts an object vertically:

• If tension equals the gravitational force, the object remains static or moves at a constant velocity.
• If tension exceeds the gravitational force, the object accelerates upwards.

The task is to calculate the maximum upward acceleration before the tension surpasses its limit and risks breaking. By understanding how tension works in tandem with gravitational force, we ensure safe and efficient use of ropes, cords, and similar tools in everyday applications.