A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1660 kg. It has strayed too close to a black hole having a mass 103 times that of the Sun (Msun = 1.99 x 1030 kg). The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. Assume that the mass of the spacecraft is evenly distributed.

(a) Determine the gravitational force on the front half of the ship. (N)

(b) Determine the gravitational force on the rear half of the ship. (N)

(c) What is the difference in the gravitational pull on the front and rear halves of the ship? This difference in gravitational pull grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart. (N)

Use Newtons law of gravity.

1.9e27,

To solve this problem, we will use Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where:
F is the gravitational force,
G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects interacting,
and r is the distance between the centers of the two objects.

(a) Determine the gravitational force on the front half of the ship:

First, let's calculate the mass of the front half of the ship.
Given:
Length of the ship (L) = 100 m
Mass with occupants (M) = 1660 kg

Since the mass is evenly distributed, the mass of the front half is:
Mass of the front half (m_front) = M / 2 = 1660 kg / 2 = 830 kg

Now, let's calculate the distance between the front half of the ship and the center of the black hole.
Given:
Distance from nose to black hole center (r_front) = 10.0 km = 10,000 m

Using Newton's law of universal gravitation, we can find the gravitational force on the front half of the ship:

F_front = G * ((m_front * m_blackhole) / r_front^2)

Given:
Mass of black hole (M_blackhole) = 103 * Msun = 103 * 1.99 x 10^30 kg

Plugging in the values, we get:

F_front = 6.67430 × 10^-11 N m^2/kg^2 * ((830 kg * (103 * 1.99 x 10^30 kg)) / (10,000 m)^2)

Calculating this expression will give us the gravitational force on the front half of the ship.

(b) Determine the gravitational force on the rear half of the ship:

To find the gravitational force on the rear half of the ship, we need to calculate the distance between the rear half of the ship and the center of the black hole. Since the ship's length is 100 m, and the distance from the nose to the center of the black hole is 10,000 m, we can deduce that the distance between the rear half of the ship and the center of the black hole is:

Distance from rear to black hole center (r_rear) = r_front + Length of the ship (L) / 2
= 10,000 m + 100 m / 2
= 10,000 m + 50 m
= 10,050 m

Now, using Newton's law of universal gravitation, we can find the gravitational force on the rear half of the ship:

F_rear = G * ((m_rear * m_blackhole) / r_rear^2)

Like before, the mass of the rear half is the same as the front half, so:

Mass of the rear half (m_rear) = 830 kg

Plugging in the values, we get:

F_rear = 6.67430 × 10^-11 N m^2/kg^2 * ((830 kg * (103 * 1.99 x 10^30 kg)) / (10,050 m)^2)

Calculating this expression will give us the gravitational force on the rear half of the ship.

(c) What is the difference in the gravitational pull on the front and rear halves of the ship?

To find the difference in gravitational pull on the front and rear halves of the ship, we subtract the force on the rear half from the force on the front half:

Difference in gravitational pull = F_front - F_rear

Calculating this expression will give us the difference in gravitational pull between the front and rear halves of the ship.

To find the gravitational force on the front half and rear half of the spacecraft, we can use Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where:
F is the gravitational force,
G is the gravitational constant (6.674 x 10^-11 N*m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between their centers of mass.

(a) To calculate the gravitational force on the front half of the ship, we need to find the mass of the front half and the distance between the front half and the black hole.

The mass of the front half of the spacecraft can be determined by dividing the total mass (including occupants) evenly between the front and rear halves:

mass_front = total_mass / 2
= 1660 kg / 2
= 830 kg

The distance between the front half of the spacecraft and the center of the black hole is given as 10.0 km. However, we need to convert it to meters:

distance_front = 10.0 km * 1000
= 10000 m

Now we can calculate the gravitational force on the front half:

F_front = G * (mass_front * mass_black_hole) / r^2

Substituting the given values:

F_front = (6.674 x 10^-11 N*m^2/kg^2) * (830 kg * (103 * Msun))
/ (10000 m)^2

Where Msun is the mass of the Sun (1.99 x 10^30 kg).

(b) Similarly, to calculate the gravitational force on the rear half of the ship, we use the same formula:

mass_rear = total_mass / 2
= 1660 kg / 2
= 830 kg

The distance between the rear half of the spacecraft and the center of the black hole is also 10.0 km, which we convert to meters:

distance_rear = 10.0 km * 1000
= 10000 m

Now we can calculate the gravitational force on the rear half:

F_rear = G * (mass_rear * mass_black_hole) / r^2

Substituting the given values:

F_rear = (6.674 x 10^-11 N*m^2/kg^2) * (830 kg * (103 * Msun))
/ (10000 m)^2

(c) To find the difference in gravitational pull on the front and rear halves of the ship, we subtract the force on the rear half from the force on the front half:

Difference = F_front - F_rear

Substituting the values obtained in the previous calculations, we can find the difference in gravitational pull.

It's important to note that as the spacecraft gets closer to the black hole, the gravitational forces increase significantly, putting the body of the ship under extreme tension. Eventually, these forces can tear the ship apart.