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Chapter 4: Problem 5

Consider a \(2.0-\mathrm{kg}\) ball and a \(6.0-\mathrm{kg}\) ball in free fall. (a) What is the net force acting on each? (b) What is the acceleration of each?

Short Answer

Expert verified
(a) Ball A: 19.6 N, Ball B: 58.8 N. (b) Both accelerations: 9.8 m/s虏.

Step by step solution

01

Understand the Problem

We have two balls, one with mass 2.0 kg (Ball A) and another with mass 6.0 kg (Ball B), and we need to find the net force and acceleration for each ball during free fall. Remember that free fall means only the force of gravity acts on them.
02

Recall the Formula for Force

The formula for calculating the force due to gravity, which is also the net force in free fall, is given by Newton's second law: \[ F = m imes g \] where \( F \) is the force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \ \text{m/s}^2 \) on Earth.
03

Calculate the Net Force for Ball A

Using the formula \( F = m \times g \): \[ F_A = 2.0 \ \text{kg} \times 9.8 \ \text{m/s}^2 = 19.6 \ \text{N} \]The net force acting on Ball A is 19.6 Newtons.
04

Calculate the Net Force for Ball B

Similarly, for Ball B:\[ F_B = 6.0 \ \text{kg} \times 9.8 \ \text{m/s}^2 = 58.8 \ \text{N} \]The net force acting on Ball B is 58.8 Newtons.
05

Understand Acceleration in Free Fall

In free fall, the only force acting on an object is gravity, thus the acceleration for both objects is equal to the acceleration due to gravity \( g \). Therefore,
06

State the Acceleration for Each Ball

The acceleration for both Ball A and Ball B can be stated as: \[ a = 9.8 \ \text{m/s}^2 \]This is because the acceleration due to gravity is the same for all objects in free fall, regardless of their masses.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free Fall
In physics, free fall refers to the motion of an object when it is falling solely under the influence of gravity. This means no other forces, like air resistance, are acting on it. During free fall, an object will continue to accelerate downwards due to the force of gravity. On Earth, this happens at a constant rate defined by the acceleration due to gravity, which is approximately 9.8 m/s虏. Whether it's a feather or a ball, in a vacuum, they will fall at the same rate because only gravity affects their motion. This concept helps us understand that in free fall, every object experiences a force equal to their mass times the acceleration due to gravity.
Net Force
The net force acting on an object is the sum of all the forces acting upon it. In the context of free fall, the only force to consider is the gravitational force. For an object in free fall, the net force is equal to the gravitational force, calculated as the object's mass multiplied by the acceleration due to gravity (9.8 m/s虏 on Earth). This makes the calculation straightforward:
  • For a 2.0 kg ball: Net force is 19.6 N.
  • For a 6.0 kg ball: Net force is 58.8 N.
The net force is critical because it determines the motion of the object according to Newton's Second Law.
Acceleration due to Gravity
Acceleration due to gravity is a constant value that represents how fast an object's velocity changes as it falls, under the influence of the Earth's gravitational pull. This value, approximately 9.8 m/s虏, indicates that with every second of free fall, the object鈥檚 speed increases by 9.8 meters per second.
  • This acceleration is the same for every object, regardless of its mass.
  • It represents a fundamental force that acts on all masses within the Earth's gravitational field.
This constant acceleration allows us to predict the dynamics of an object in free fall accurately.
Mass and Weight
Mass and weight are related but distinct concepts in physics. Mass is a measure of how much matter an object contains and is constant regardless of location. Its unit is kilograms (kg). On the other hand, weight is the force exerted by gravity on that mass. On Earth, weight can be calculated using the formula: weight = mass 脳 gravitational acceleration (9.8 m/s虏).
  • Thus, a 2.0 kg ball has a weight of 19.6 N.
  • A 6.0 kg ball has a weight of 58.8 N.
Weight changes with the strength of the gravitational field, but mass remains unchanged.
Newton's Second Law
Newton's Second Law of Motion is a cornerstone in understanding how objects move under the influence of forces. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). This law explains why heavier objects require more force to achieve the same acceleration as lighter objects. In free fall, the acceleration is due to gravity alone, so each object's net force is its weight.
  • For example, for a 2.0 kg ball, the force is 2.0 kg 脳 9.8 m/s虏 = 19.6 N.
  • Similarly, for a 6.0 kg ball, the force is 6.0 kg 脳 9.8 m/s虏 = 58.8 N.
This illustrates that while the force and weight increase with mass, the free fall acceleration remains constant.

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Most popular questions from this chapter

A 75.0 -kg person is standing on a scale in an elevator. What is the reading of the scale in newtons if the elevator is (a) at rest, (b) moving up at a constant velocity of \(2.00 \mathrm{~m} / \mathrm{s},\) and \((\mathrm{c})\) accelerating up at \(2.00 \mathrm{~m} / \mathrm{s}^{2} ?\)

A book is sitting on a horizontal surface. (a) There is (are) (1) one, (2) two, or (3) three force(s) acting on the book. (b) Identify the reaction force to each force on the book.

The weight of a 500 -kg object is 4900 N. (a) When the object is on a moving elevator, its measured weight could be (1) zero, (2) between zero and \(4900 \mathrm{~N}\), (3) more than \(4900 \mathrm{~N},\) (4) all of the preceding. Why? (b) Describe the motion if the object's measured weight is only \(4000 \mathrm{~N}\) in a moving elevator.

IE oo (a) You are told that an object has zero acceleration. Which of the following is true: (1) The object is at rest; (2) the object is moving with constant velocity; (3) either (1) or (2) is possible; or (4) neither 1 nor 2 is possible. (b) Two forces on the object are \(F_{1}=3.6 \mathrm{~N}\) at \(74^{\circ}\) below the \(+x\) -axis and \(F_{2}=3.6 \mathrm{~N}\) at \(34^{\circ}\) above the \(-x\) -axis. Is there a third force on the object? Why or why not? If there is a third force, what is it?

A loaded Boeing 747 jumbo jet has a mass of \(2.0 \times 10^{5} \mathrm{~kg} .\) What net force is required to give the plane an acceleration of \(3.5 \mathrm{~m} / \mathrm{s}^{2}\) down the runway for takeoffs?
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