Chapter 4: Problem 24
A student of mass 45 kg jumps off a high diving board. What is the acceleration of the earth toward her as she accelerates toward the earth with an acceleration of 9.8 m/s\(^2\)? Use 6.0 \(\times\) 10\(^{24}\) kg for the mass of the earth, and assume that the net force on the earth is the force of gravity she exerts on it.
Short Answer
Step by step solution
Understand Newton's Third Law
Calculate the Gravitational Force Exerted by the Student
Apply Newton's Second Law to the Earth
Calculate the Earth's Acceleration
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gravitational Force
The strength of gravitational force depends on two key factors:
- Mass of the objects involved: Larger masses exert a stronger gravitational pull.
- Distance between the centers of the two objects: Greater distances result in weaker gravitational forces.
Acceleration
Newton's Second Law of Motion connects acceleration (\( a \)) with force (\( F \)) and mass (\( m \)) using the formula: \[ F = m \cdot a \]This formula implies that:
- The acceleration of an object is directly proportional to the force acting upon it.
- Acceleration is inversely proportional to the mass of the object.
The relationship between acceleration and the forces acting on objects is crucial in understanding motion, and it is especially pivotal when calculating the Earth's slight acceleration in response to forces applied to it.
Mass of the Earth
An important point to note is that the Earth's enormous mass results in very minimal acceleration when subjected to normal forces. This concept was demonstrated in the exercise example, where the planet's acceleration towards a student jumping off a diving board was almost negligible: \( 7.35 \times 10^{-23} \text{ m/s}^2 \).
- Such a tiny acceleration is due to the Earth’s vast mass compared to the small external force applied by an individual student.
- Therefore, while any force applied has a theoretical impact, the Earth's movement is virtually imperceptible due to its hugeness.