A uniform rod of mass \(m_{\text {rod }}\) and length \(\ell_{\text {rod }}\) lies along an \(x\) axis far from any stars or planets, with the center of the rod at the origin (Figure P13.50). A ball of mass \(m_{\text {ball }}\) is located at position \(x_{\text {ball }}\) on the axis.

(a) Write an expression for the gravitational potential energy of the system made up of the ball and a small element \(d m_{\text {rod }}\) of the rod located at position \(x_{d m}\).

(b) Integrate your expression over the length of the rod to determine the potential energy of the system for \(x_{\text {ball }}>\ell_{\text {rod }} / 2\).

(c) It can be shown that for such a system the gravitational force exerted by the rod on an object located a distance \(x\) from the origin is \(F_{x}=-d U / d x\). Using this relationship, compute the force exerted by the rod on the ball at any distance \(x\) from the rod center.

Data from Figure P13.50

Transcribed Image Text:

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