Multiple Choice
a) Set up the integral for the gravitational force between a rod with a density λ and length 2L, and a mass m at an arbitrary distance D directly above the midpoint of the rod.
b) Evaluate the integral.
8. Centripetal Forces & Gravitation
Mass Distribution with Calculus
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Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
In the context of mass distribution, how can calculus be used to determine the center of mass of an object?
A
By differentiating the mass density function with respect to time.
B
By integrating the mass density function over the volume of the object.
C
By using algebraic equations to solve for mass distribution.
D
By calculating the average velocity of the object.
Verified step by step guidance1
Understand that the center of mass is a point representing the mean position of the mass in a body or system.
Recognize that for a continuous mass distribution, the center of mass can be found using calculus by integrating the mass density function over the volume of the object.
Set up the integral for the center of mass. For a three-dimensional object, the center of mass coordinates (x_cm, y_cm, z_cm) can be found using the integrals: x_cm = (1/M) ∫ x * ρ(x, y, z) dV, y_cm = (1/M) ∫ y * ρ(x, y, z) dV, z_cm = (1/M) ∫ z * ρ(x, y, z) dV, where ρ(x, y, z) is the mass density function and M is the total mass.
Calculate the total mass M of the object by integrating the mass density function over the entire volume: M = ∫ ρ(x, y, z) dV.
Evaluate the integrals for x_cm, y_cm, and z_cm using the calculated total mass M to find the coordinates of the center of mass.
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