8. Centripetal Forces & Gravitation

In the context of mass distribution with calculus, which of the following physical quantities is directly affected by the distribution of mass in an object?

Given three groups of particles located along the x-axis: Group A consists of 5 particles each of mass $2$ kg at positions $0$ m, $1$ m, $2$ m, $3$ m, and $4$ m; Group B consists of 4 particles each of mass $3$ kg at positions $0$ m, $2$ m, $4$ m, and $6$ m; Group C consists of 6 particles each of mass $1.5$ kg at positions $0$ m, $1$ m, $2$ m, $3$ m, $4$ m, and $5$ m. Which group has the greatest total mass?

Two cylinders, A and B, are placed on a beam as shown in Figure 1. Cylinder A has a mass of $8kg$ and cylinder B has a mass of $6kg$. If the beam is $2m$ long and cylinder A is placed at $0.5m$ from the left end while cylinder B is placed at $1.5m$ from the left end, what is the position of the center of mass of the system measured from the left end of the beam?

A golf ball has more $mass$ than a tennis ball because it

In the context of mass distribution and calculus, which statement correctly describes the relationship between $mass$ and $weight$?

A thin rod of length $L$ lies along the x-axis from $x=0$ to $x=L$. Its linear mass density varies with position as $\lambda \left(x\right)=\alpha x$, where $\alpha$ is a constant. What is the total mass of the rod?

A thin rod of length $L$ has a linear mass density given by $\lambda ={\lambda }_0(1+\alpha x)$, where $x$ is the distance from one end, ${\lambda }_0$ and $\alpha$ are constants. What is the total mass of the rod?

In the context of mass distribution, how can calculus be used to determine the $center of mass$ of an object?

In the context of mass distribution and calculus, what is the most important thing to remember about the relationship between mass and weight?

Given two rods of equal length $L$, one with a uniform linear mass density $\rho$ and the other with a linear mass density that varies as $\rho =\rho 0_x$ from one end to the other, which rod has more total mass?

Which of the following is a measurement of an object's mass?

In the context of mass distribution with calculus, if a player has more $m$, they will also have more which of the following physical quantities, assuming all other factors remain constant?

In the context of mass distribution with calculus, your vehicle's $m(x)$ affects its $I$.

If two different objects have different mass distributions described by their respective density functions, would a dozen of each object necessarily have the same total mass?

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.