/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 A raindrop has a mass of \(5.2 \... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 23

A raindrop has a mass of \(5.2 \times 10^{-7} \mathrm{kg}\) and is falling near the surface of the earth. Calculate the magnitude of the gravitational force exerted (a) on the raindrop by the earth and (b) on the earth by the raindrop.

Short Answer

Expert verified
(a) The force on the raindrop is \(5.1 \times 10^{-6} \text{ N}\). (b) The force on the Earth is also \(5.1 \times 10^{-6} \text{ N}\).

Step by step solution

01

Identifying Known Values

We are given the mass of the raindrop as \(m = 5.2 \times 10^{-7} \text{ kg}\). The acceleration due to gravity near the Earth's surface is \(g = 9.81 \text{ m/s}^2\).
02

Calculating the Gravitational Force on the Raindrop

Using the formula for gravitational force, \(F = mg\), where \(m\) is the mass and \(g\) is the acceleration due to gravity, we substitute the values: \[ F = (5.2 \times 10^{-7} \text{ kg})(9.81 \text{ m/s}^2) \].
03

Performing the Calculation

Calculate \(F\): \[ F = 5.2 \times 10^{-7} \times 9.81 = 5.1012 \times 10^{-6} \text{ N} \].Thus, the gravitational force exerted on the raindrop by the Earth is approximately \(5.1 \times 10^{-6} \text{ N}\).
04

Understanding Action-Reaction Pairs

According to Newton's Third Law of Motion, the gravitational force exerted by the raindrop on the Earth is equal in magnitude and opposite in direction to the force exerted by the Earth on the raindrop.
05

Identifying the Force on the Earth

The force exerted on the Earth by the raindrop is also \(5.1 \times 10^{-6} \text{ N}\), directed upwards.

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Key Concepts

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

Raindrop Mass
Here, we discuss the concept of mass specific to a raindrop. A raindrop may seem insignificant due to its small size, but its mass plays a crucial role in physical calculations. The mass of the raindrop in this particular exercise is given as a very minuscule value: \(5.2 \times 10^{-7} \text{ kg}\). This measurement represents how much matter is contained in the raindrop, which is essential for calculating its interaction with Earth's gravitational field.
  • The mass of a typical raindrop falls within the range of such small numbers, usually in the range of micrograms to milligrams.
  • Mass is a scalar quantity, meaning it has magnitude but no direction.
  • Identifying the mass accurately is vital for solving problems involving forces, specifically gravitational force in this case.
Calculations involving such small masses underscore physics' precision and our ability to predict interactions at very tiny scales.
Newton's Third Law
Newton's Third Law of Motion is fundamental in understanding interactions between two objects. It states that for every action, there is an equal and opposite reaction. This is evident when we consider the gravitational interaction between a raindrop and the Earth.
  • When the Earth exerts a gravitational force on the raindrop, the raindrop simultaneously exerts a force back on the Earth.
  • The forces in this action-reaction pair are equal in magnitude. Hence, if the Earth pulls on the raindrop with a force of \(5.1 \times 10^{-6} \text{ N}\), the raindrop also pulls back on the Earth with the same force.
  • Despite the magnitude being the same, the effects of these forces differ due to the massive difference in mass between the Earth and the raindrop, which affects acceleration.
Understanding this law helps clarify how gravitational forces function even at a micro-scale.
Acceleration Due to Gravity
Acceleration due to gravity is denoted by \(g\) and is approximately \(9.81 \text{ m/s}^2\) near Earth's surface. It is a critical component in calculating the gravitational force acting on objects. It represents how much an object鈥檚 velocity increases as it falls freely towards the Earth.
  • This constant acceleration applies uniformly to all objects in a vacuum, disregarding air resistance or other forces.
  • In calculations, such as finding the gravitational force on a raindrop, we multiply the object's mass by \(g\) (i.e., \(F = mg\)).
  • Despite external factors such as air resistance, everything on Earth experiences this acceleration, which profoundly impacts motion dynamics.
Understanding \(g\) is crucial for physics, as it impacts everything from free-fall equations to the understanding of orbits in space.

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Most popular questions from this chapter

A rocket of mass \(4.50 \times 10^{5} \mathrm{kg}\) is in flight. Its thrust is directed at an angle of \(55.0^{\circ}\) above the horizontal and has a magnitude of \(7.50 \times 10^{6} \mathrm{N} .\) Find the magnitude and direction of the rocket's acceleration. Give the direction as an angle above the horizontal.

A duck has a mass of \(2.5 \mathrm{kg}\). As the duck paddles, a force of \(0.10 \mathrm{N}\) acts on it in a direction due east. In addition, the current of the water exerts a force of \(0.20 \mathrm{N}\) in a direction of \(52^{\circ}\) south of east. When these forces begin to act, the velocity of the duck is \(0.11 \mathrm{m} / \mathrm{s}\) in a direction due east. Find the magnitude and direction (relative to due east) of the displacement that the duck undergoes in \(3.0 \mathrm{s}\) while the forces are acting.

At an instant when a soccer ball is in contact with the foot of a player kicking it, the horizontal or \(x\) component of the ball's acceleration is \(810 \mathrm{m} / \mathrm{s}^{2}\) and the vertical or \(y\) component of its acceleration is \(1100 \mathrm{m} / \mathrm{s}^{2} .\) The ball's mass is \(0.43 \mathrm{kg} .\) What is the magnitude of the net force acting on the soccer ball at this instant?

A 292 -kg motorcycle is accelerating up along a ramp that is inclined \(30.0^{\circ}\) above the horizontal. The propulsion force pushing the motorcycle up the ramp is \(3150 \mathrm{N}\), and air resistance produces a force of \(250 \mathrm{N}\) that opposes the motion. Find the magnitude of the motorcycle's acceleration.

A cup of coffee is on a table in an airplane flying at a constant altitude and a constant velocity. The coefficient of static friction between the cup and the table is \(0.30 .\) Suddenly, the plane accelerates forward, its altitude remaining constant. What is the maximum acceleration that the plane can have without the cup sliding backward on the table?
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