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. The setup shown by the diagram below was used to calculate the force experienced by a small mass m situated at a distance r from the geometric center of a spherical shell with mass M, radius R, and a wall thickness of t. M C de R 0 RdᎾ A Rsine B r dF $ Ф m If the mass is situated outside the spherical shell, the force can be calculated by integration from l = (r - R) to l= (r+ R), as shown here: l=r+R R r²-R² SdF - "T" Comport (1² R² ) de SdF="Gmpat- 1+ 1² l=r-R What would the limits of integration be if the mass were situated inside the spherical shell? Include a labeled diagram to represent this situation.

. The setup shown by the diagram below was used to calculate the force experienced by a small mass m situated at a distance r from the geometric center of a spherical shell with mass M, radius R, and a wall thickness of t. M C de R 0 RdᎾ A Rsine B r dF $ Ф m If the mass is situated outside the spherical shell, the force can be calculated by integration from l = (r - R) to l= (r+ R), as shown here: l=r+R R r²-R² SdF - "T" Comport (1² R² ) de SdF="Gmpat- 1+ 1² l=r-R What would the limits of integration be if the mass were situated inside the spherical shell? Include a labeled diagram to represent this situation.

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1. The setup shown by the diagram below was used to calculate the force experienced by a small mass \( m \) situated at a distance \( r \) from the geometric center of a spherical shell with mass \( M \), radius \( R \), and a wall thickness of \( t \). The diagram shows: - A circle representing the spherical shell with radius \( R \). - A small mass \( m \) located outside the spherical shell at a distance \( r \) from the center \( C \). - An angle \( \theta \) with an arc length \( Rd\theta \), and two lines forming an angle \( \phi \) with the horizontal that meet at the small mass \( m \). - A small force element \( dF \) acting on mass \( m \). - The line \( \ell \) that extends from the center \( C \) to the small mass \( m \). If the mass is situated **outside** the spherical shell, the force can be calculated by integration from \( \ell = (r - R) \) to \( \ell = (r + R) \), as shown here: \[ \int_{\ell = r-R}^{\ell = r+R} dF = \int_{\ell = r-R}^{\ell = r+R} Gm\rho\pi \frac{R}{r^2} \left( 1 + \frac{r^2 - R^2}{\ell^2} \right) d\ell \] What would the limits of integration be if the mass were situated **inside** the spherical shell? Include a labeled diagram to represent this situation.
Transcribed Image Text:1. The setup shown by the diagram below was used to calculate the force experienced by a small mass \( m \) situated at a distance \( r \) from the geometric center of a spherical shell with mass \( M \), radius \( R \), and a wall thickness of \( t \). The diagram shows: - A circle representing the spherical shell with radius \( R \). - A small mass \( m \) located outside the spherical shell at a distance \( r \) from the center \( C \). - An angle \( \theta \) with an arc length \( Rd\theta \), and two lines forming an angle \( \phi \) with the horizontal that meet at the small mass \( m \). - A small force element \( dF \) acting on mass \( m \). - The line \( \ell \) that extends from the center \( C \) to the small mass \( m \). If the mass is situated **outside** the spherical shell, the force can be calculated by integration from \( \ell = (r - R) \) to \( \ell = (r + R) \), as shown here: \[ \int_{\ell = r-R}^{\ell = r+R} dF = \int_{\ell = r-R}^{\ell = r+R} Gm\rho\pi \frac{R}{r^2} \left( 1 + \frac{r^2 - R^2}{\ell^2} \right) d\ell \] What would the limits of integration be if the mass were situated **inside** the spherical shell? Include a labeled diagram to represent this situation.
2. Using the limits of integration from Q1 (above), show that the gravitational force *within* a spherical shell with mass \( M \), radius \( R \), and a wall thickness of \( t \) is zero. Explicitly show each step. (Hint: there should be at least eight steps.)
Transcribed Image Text:2. Using the limits of integration from Q1 (above), show that the gravitational force *within* a spherical shell with mass \( M \), radius \( R \), and a wall thickness of \( t \) is zero. Explicitly show each step. (Hint: there should be at least eight steps.)
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