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8. Centripetal Forces & Gravitation
Acceleration Due to Gravity
9:46 minutes
Problem 6.61b
Textbook Question
Textbook Question(III) Two identical particles, each of mass m, are located on the x axis at x= +x₀ and x = -x₀.
(b) At what point (or points) on the y axis is the magnitude of g a maximum value, and what is its value there? [Hint: Take the derivative dg/dy .]
Verified step by step guidance
1
Identify the gravitational force exerted by each particle on a point on the y-axis. Use the formula for gravitational force, F = G \frac{m_1 m_2}{r^2}, where G is the gravitational constant, m_1 and m_2 are the masses of the particles, and r is the distance between the masses.
Express the distance r from each particle to a point y on the y-axis using the Pythagorean theorem. Since the particles are at positions (+x_0, 0) and (-x_0, 0), the distance to a point (0, y) on the y-axis from either particle is r = \sqrt{x_0^2 + y^2}.
Write the expression for the gravitational field g at point y on the y-axis due to each particle. The gravitational field g due to a mass m at a distance r is given by g = G \frac{m}{r^2}. Since the particles are identical and their contributions add up vectorially along the y-axis, the total gravitational field at point y is g = 2G \frac{m}{(x_0^2 + y^2)^{3/2}} y.
To find the point where the magnitude of g is maximum, take the derivative of the magnitude of g with respect to y, set it to zero, and solve for y. This involves differentiating g = 2G \frac{m}{(x_0^2 + y^2)^{3/2}} y with respect to y.
Solve the equation obtained from the derivative to find the value(s) of y that maximize the magnitude of g. Check the second derivative or use boundary conditions to confirm that these points indeed correspond to a maximum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Gravitational Field Strength (g)
Gravitational field strength, denoted as g, is a vector quantity that represents the force experienced by a unit mass in a gravitational field. It is influenced by the masses of the objects creating the field and their distances from the point of interest. In this scenario, the gravitational field strength due to two identical particles can be calculated using Newton's law of gravitation, which states that the force between two masses decreases with the square of the distance between them.
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Derivative and Optimization
The derivative of a function measures how the function's output changes as its input changes. In the context of this problem, taking the derivative of the gravitational field strength with respect to the y-coordinate allows us to find the points where g is maximized. By setting the derivative equal to zero, we can identify critical points that may correspond to maximum or minimum values of g, which is essential for solving the problem.
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Symmetry in Physics
Symmetry plays a crucial role in simplifying physical problems. In this case, the identical particles are symmetrically placed along the x-axis, which implies that the gravitational field strength on the y-axis will also exhibit symmetry. This symmetry can help in determining the points where the gravitational field strength is maximized, as the contributions from both particles will be equal and additive at certain locations along the y-axis.
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