Answered

Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

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.
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 ln(1+x).)
b) Use Fx=−dU/dx to find the magnitude of the gravitational force exerted on the sphere by the rod.

Sagot :

ajeigbeibraheem

Answer:

Explanation:

From the given information:

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 dr of the rod can be computed as:

[tex]dU = -\dfrac{GMm}{L}*\dfrac{dr}{r}[/tex]

where,

G = gravitational constant

[tex]U = - \int^{x+L}_{x}Gm\dfrac{M}{L}*\dfrac{dr}{r}[/tex]

[tex]U = - \dfrac{GMm}{L}\int^{x+L}_{x}\dfrac{dr}{r}[/tex]

By taking the integral within the limit

[tex]U = - \dfrac{GMm}{L} \Big[In \ r\Big]^{x+L}_{x}[/tex]

[tex]\mathbf{\implies - \dfrac{GmM}{L} In \Big(\dfrac{{x+L}}{{x}}\Big)}[/tex]

b)

By using [tex]F= -\dfrac{dU}{dx}[/tex], the magnitude of the gravitational force can be determined as follows:

Here, we have:

[tex]F = -\dfrac{d}{dx} \Big [\dfrac{-GmM}{L}In(\dfrac{x+l}{x}) \Big ] \\ \\ = \dfrac{GmM}{L}\times \dfrac{x}{x+L}\times (0-\dfrac{L}{x^2}) \\ \\ By \ solving \\ \\ \mathbf{ =-\dfrac{GmM}{x(x+L)}}[/tex]

From above, the negative sign indicates an attractive force

Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.