91Ó°ÊÓ

Two skaters, a man and a woman, are standing on ice. Neglect any friction between the skate blades and the ice. The woman pushes on the man with a certain force that is parallel to the ground. (a) Must the man accelerate under the action of this force? If so, what three factors determine the magnitude and direction of his acceleration? (b) Is there a corresponding force exerted on the woman? If so, where does it originate? Is this force related to the magnitude and direction of the force the woman exerts on the man? If so, how? The mass of the man is \(82 \mathrm{~kg}\) and that of the woman is \(48 \mathrm{~kg}\). The woman pushes on the man with a force of \(45 \mathrm{~N}\) due east. Determine the acceleration (magnitude and direction) of (a) the man and (b) the woman.

Step by step solution

01

Understanding Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This means if the woman exerts a force on the man, the man exerts an equal force in the opposite direction back on the woman.

02

Factors Affecting Man's Acceleration

The man's acceleration is determined by Newton's Second Law, which states that acceleration is the net force divided by the mass of the object. Thus, the factors are: 1) the amount of force applied, 2) the man's mass, and 3) the direction of the force applied.

03

Analyzing Force Interactions

When the woman pushes the man, he must accelerate due to the external force exerted on him. Simultaneously, by Newton's Third Law, the woman experiences an equal and opposite force exerted by the man.

04

Calculating Man's Acceleration

To find the man's acceleration, apply Newton’s Second Law: \( a = \frac{F}{m} \).- Given: \( F = 45 \mathrm{~N} \) and \( m = 82 \mathrm{~kg} \).- Calculation: \[ a = \frac{45 \mathrm{~N}}{82 \mathrm{~kg}} = 0.5488 \mathrm{~m/s^2} \]The man accelerates at \(0.5488 \mathrm{~m/s^2}\) east.

05

Calculating Woman's Acceleration

Similarly, to find the woman's acceleration, use: \( a = \frac{F}{m} \).- Given: \( F = 45 \mathrm{~N} \) and \( m = 48 \mathrm{~kg} \).- Calculation:\[ a = \frac{45 \mathrm{~N}}{48 \mathrm{~kg}} = 0.9375 \mathrm{~m/s^2} \]The woman accelerates at \(0.9375 \mathrm{~m/s^2}\) west.

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Key Concepts

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

Newton's Third Law

Newton's Third Law of Motion is a fundamental principle in physics that helps us understand how objects interact. It states that for every action, there is an equal and opposite reaction. In simpler terms, whenever one object exerts a force on another object, the second object exerts an equally strong force in the opposite direction on the first object. This is crucial in understanding how forces are not isolated but rather part of a mutual interaction.
In the case of the skaters, when the woman pushes on the man with a force, the man exerts an equal force in the opposite direction on the woman. This pair of forces is what helps us predict movements, as they occur simultaneously and with equal magnitude.

Newton's Second Law

Newton's Second Law of Motion is another cornerstone of physics that describes how the velocity of an object changes when it is subjected to external forces. According to this law, the acceleration of an object is directly proportional to the net external force acting upon it and is inversely proportional to its mass. This can be mathematically expressed as:

• \( F = ma \) or \( a = \frac{F}{m} \)

This means that if you know the total force exerted on an object and its mass, you can easily find its acceleration.
In the exercise, we apply this law to find how the man and the woman will accelerate due to the forces they exert on each other. The force is known, and the mass of each person helps in determining their respective accelerations.

Force Interactions

Force interactions occur when two objects exert forces on each other, leading to motion. Understanding these interactions is key to predicting how objects move in response to each other's forces. When considering the interaction between the skater man and woman, the forces they exert are part of a pair, aligning with Newton's Third Law.
As the woman pushes the man, her action results in the man accelerating. According to Newton's Third Law, the man exerts an equal force back on the woman. Hence, this interaction not only explains the man's motion but also ensures the woman experiences a force in the opposite direction.
In essence, no force exists in isolation; there is always another force that is equal in magnitude but opposite in direction involved in every interaction.

Acceleration Calculation

Calculating acceleration involves using Newton’s Second Law by dividing the net force by the mass of the object:

• For the man's acceleration: Using \( F = 45 \mathrm{~N} \) and \( m = 82 \mathrm{~kg} \), the formula \( a = \frac{F}{m} \) gives:
• \[ a = \frac{45 \mathrm{~N}}{82 \mathrm{~kg}} = 0.5488 \mathrm{~m/s^2} \]

The man's acceleration is therefore eastward at \( 0.5488 \mathrm{~m/s^2} \).

• For the woman's acceleration: With \( F = 45 \mathrm{~N} \) and \( m = 48 \mathrm{~kg} \), we use the same calculation method:
• \[ a = \frac{45 \mathrm{~N}}{48 \mathrm{~kg}} = 0.9375 \mathrm{~m/s^2} \]

The woman accelerates westward at \( 0.9375 \mathrm{~m/s^2} \).
Through these calculations, we can determine how each skater moves in response to the forces exerted during their interaction.