A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod (Fig. E13.34). Calculate the gravitational potential energy of the rod–sphere system. Take the potential energy to be zero when the rod and sphere are infinitely far apart. Show that your answer reduces to the expected result when x is much larger than L.

Consider the ringshaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use F_x = -dU/dx to find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and F_x when x = 0? Explain why these results make sense.

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

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?

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

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?