Problem 4 - Astronomy Journey to the Cosmic Frontier

Question

Two forces, oppositely directed, act on a body. The force acting toward the right is twice as strong as the force acting toward the left. Describe the motion of the body.

Step-by-Step Solution

Answer

The body will move to the right due to the stronger force.

Step 1: Identify the Forces

Let's denote the force acting toward the left as \( F_1 \) and the force acting toward the right as \( F_2 \). According to the problem, we are given that the force toward the right is twice as strong as the force toward the left. So, \( F_2 = 2F_1 \).

Step 2: Apply Newton's Second Law

Newton's Second Law states that the motion of a body is determined by the net force acting on it, which is the sum of all the forces. In this case, the net force \( F_{net} \) can be calculated as the difference between \( F_2 \) and \( F_1 \) since they are oppositely directed. Thus, \( F_{net} = F_2 - F_1 \). Substituting the relation we know, \( F_{net} = 2F_1 - F_1 = F_1 \).

Step 3: Determine the Direction of Motion

Since the net force \( F_{net} = F_1 \) is positive (since we defined \( F_2 \) as being greater and to the right), the resultant force is directed toward the right. According to Newton's second law, this means the body will accelerate in the direction of the net force, which is to the right in this case.

Key Concepts

Net Force Calculation

In physics, understanding how forces interact is essential. Taking Newton's Second Law into account, we know that the net force acting on a body dictates its motion. The net force is the total of all the forces when they are combined. This is not simply adding magnitudes; direction matters greatly. In problems where forces are opposing, like in our exercise, calculation is straightforward.
To find the net force, we subtract the smaller force (moving left) from the larger force (moving right). This gives us the net force. For example, with forces denoted as \( F_1 \) and \( F_2 \), where \( F_2 = 2F_1 \), the net force is \( F_{net} = F_2 - F_1 \). Simple arithmetic tells us this equals \( F_1 \).
Thus, correctly calculating the net force is crucial for predicting how an object will behave under these influences. It shows how unbalanced forces lead to motion or acceleration.

Direction of Motion

The direction of motion is determined by the direction of the net force. Once the net force has been calculated, understanding its direction tells us how the object will move. Here, since the force toward the right is dominant as \( F_2 \) is twice \( F_1 \), the net force is directed to the right as well.

• The object accelerates in the direction of the greater force.
• This establishes that the object will move rightward based on the given forces.

Newton's Second Law ties net force and acceleration together, emphasizing that the larger force dictates motion. Therefore, objects move along the path of least resistance, following the dominant force's direction.

Force Equilibrium

Equilibrium of forces is an important concept in understanding motion. However, in the context of the exercise, the forces are not in equilibrium since one force is greater, leading to motion.

• Equilibrium occurs when all forces are balanced, resulting in no net force. In this state, an object remains at rest or moves at a constant velocity.
• Our case has a net force of \( F_1 \), demonstrating disequilibrium, resulting in acceleration.

While force equilibrium means stability and no change in motion, any imbalance, such as the one we have here, causes acceleration and a change in velocity. Understanding this concept helps clarify why the body in the exercise doesn't remain still.