Ch. 06 - Gravitation and Newton's Synthesis

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 06 - Gravitation and Newton's Synthesis

Problem 10

Chapter 6, Problem 10

Four masses are arranged as shown in Fig. 6–28. Determine the x and y components of the gravitational force on the mass at the origin (m). Write the force in vector notation (î, ĵ).

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Identify the masses and their positions relative to the origin. Assign coordinates to each mass (e.g., m1, m2, m3, m4) based on the figure. Assume the origin is at (0, 0) and the masses are located at specific coordinates (x, y).

Use Newton's law of gravitation to calculate the gravitational force between the mass at the origin (m) and each of the other masses. The formula is:

F = \frac{G m m_{i}}{r^{2}}

, where G is the gravitational constant, m is the mass at the origin, mi is the mass of the other object, and r is the distance between the two masses.

Break each gravitational force into its x and y components. For a mass located at (xi, yi), the x-component of the force is:

F_{x} = F_{i} \cdot \frac{x}{r}

, and the y-component is:

F_{y} = F_{i} \cdot \frac{y}{r}

, where r is the distance between the origin and the mass.

Sum the x-components and y-components of the forces from all the masses to find the net gravitational force in the x and y directions. Use:

F_{x} = \sum F_{x}

and

F_{y} = \sum F_{y}

Express the net gravitational force in vector notation as:

F = F_{x} i + F_{y} j

. This is the final representation of the gravitational force on the mass at the origin.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers. This force acts along the line connecting the centers of the two masses and is crucial for understanding how objects interact under gravity.

Vector Components

Vectors are quantities that have both magnitude and direction. In physics, forces are often represented as vectors, which can be broken down into components along the x and y axes. This decomposition allows for easier calculations and analysis of forces acting in different directions, particularly when multiple forces are involved, as in the case of the masses in the problem.

Equilibrium and Net Force

An object is in equilibrium when the net force acting on it is zero, meaning all the forces balance out. In the context of the problem, calculating the gravitational forces acting on the mass at the origin involves summing the forces from the other masses and determining their resultant vector. Understanding equilibrium helps in analyzing whether the mass will remain stationary or move due to the net force.

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Related Practice

Textbook Question

Use the binomial expansion ${(1 \pm x)}^{n} = 1 \pm n x + \frac{n (n - 1)}{2} x^{2} \pm \dots$ to show that the value of g is altered by approximately $Δ g \approx - 2 g \frac{Δ r}{r_{E}}$ at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E.

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Textbook Question

The value of g is altered by approximately $$\Delta$ g$\thickapprox$-2g$\frac{\Delta r}{r_{E}$}$ at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E. What is the meaning of the minus sign in this relation?

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Textbook Question

An inclined plane, fixed to the inside of an elevator, makes a 38° angle with the floor. A mass m slides on the plane without friction. What is its acceleration relative to the plane if the elevator accelerates downward at 0.50 g?

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