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A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance from one end of the rod, along the axis of the rod (Figure 1). Part A 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. (Hint: Use the power series expansion for In(1+z).) Express your answer in terms of m, M, L, 2, G. IVO AE ? U= Submit Request Answer Part B Use F1 = -dU/dx to find the magnitude of the gravitational force exerted on the sphere by the rod. Express your answer in terms of m, M, L, I, G. IVO AED ? F= Submit Request Answer Figure < 1 of 1 Part C Find the direction of the gravitational force exerted on the sphere by the rod. O the force is directed to the rod O the force is directed out of the rod M 17 Submit Request Answer A satellite with mass 898 kg is in a circular orbit with an orbital speed of 9240 m/s around the earth, Part A What is the new orbital speed after friction from the earth's upper atmosphere has done - 7.5 x 10 J of work on the satellite? Express your answer with the appropriate units. μΑ ? Value Units Submit Request Answer

Respuesta :

The gravitational potential energy of the rod-sphere system is - (GMm) / L X ln[ (x+l)/x ]. The magnitude of the gravitational force exerted on the sphere by the rod is - (GMm) / x(x+L) and the direction of the gravitational force exerted on the sphere by the rod is towards it.

(a) Let's have an imaginary view of the rod located at a given distance r from he the mass (m) of the sphere.

Then the equation for the potential energy as related to the small area of the of the rod can be computed as;

= dU = - (GMm)/ L X (dr/r)

where,

G = gravitational constant

Now, integrating this with the limits of x to (x + L), we get

= U = - (GMm) / L X ln[ (x+l)/x ]

(b) By using F = -dU/dt, the magnitude of the gravitational force can be determined as follows:

Here we have,

= F = - d {- (GMm) / L X ln[ (x+l)/x ]} / dt

= F = - (GMm) / x(x+L)

From above, the negative sign indicates an attractive force.

(c) As, the negative sign indicates an attractive force. Thus, the direction of the gravitational force exerted on the sphere by the rod is towards it.

To know more about Gravitational Potential Energy:

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