Suppose a spherical planet P has a uniform density of 4.96*10^3 kg/m^3 and a radius R=1904 km. For a parcel that weighs 73 N on the surface of the (spherical) earth, what is A) gp/ge and B) its weight on the surface of P? Using the "tunnel mail" system what is C) the delivery time in seconds to the opposite side of P? and D) the maximum velocity of the parcel in the tunnel? E) What minimum velocity must be given to the parcel to escape from the surface of P? At an orbit radius of 4.96R, what would be the parcel's F) potential energy and G) total energy? H) How long in s would it take the parcel to revolve about P in this orbit?
Share:
by email on facebook on twitter
To find the answers to these questions, we will need to use several formulas and concepts from physics. Let's break down each question and solve them step by step:
A) To find the ratio of the acceleration of gravity on planet P (gp) to that on Earth (ge), we can use the formula:
gp/ge = (Mp/M) * (R/Rp)^2
Where:
- Mp is the mass of planet P
- M is the mass of the Earth
- R is the radius of the Earth
- Rp is the radius of planet P
We are given the radius of planet P (R) as 1904 km. The mass of the Earth (M) is approximately 5.97*10^24 kg. To find the mass of planet P (Mp), we can use the formula:
Mp = (4/3) * π * ρ * Rp^3
Where:
- π is a constant (approximately 3.14)
- ρ is the density of planet P (given as 4.96*10^3 kg/m^3)
By substituting the values into the formulas, we can calculate the ratio of gp/ge.
B) The weight of the parcel on the surface of planet P can be calculated using the formula:
Weight = mass * gp
Where:
- mass is the mass of the parcel (given by weight divided by acceleration due to gravity on Earth, which is approximately 9.8 m/s^2)
- gp is the acceleration due to gravity on planet P (found in question A)
By substituting the values into the formula, we can calculate the weight of the parcel on the surface of planet P.
C) To calculate the delivery time of the parcel to the opposite side of planet P using the "tunnel mail" system, we need to consider the distance the parcel needs to travel and the speed of sound inside the planet.
D) The maximum velocity of the parcel in the tunnel is determined by the escape velocity at the surface of planet P.
E) The minimum velocity required to escape from the surface of planet P can be calculated using the escape velocity formula:
v_escape = √(2 * G * Mp / Rp)
Where:
- G is the gravitational constant (approximately 6.67430 * 10^-11 N*m^2/kg^2)
- Mp is the mass of planet P (found in question A)
- Rp is the radius of planet P (given)
By substituting the values into the formula, we can calculate the minimum velocity required to escape.
F) The potential energy of the parcel at an orbit radius of 4.96R can be calculated using the formula:
Potential energy = - (G * Mp * mass) / (orbit radius)
Where:
- G is the gravitational constant
- Mp is the mass of planet P (found in question A)
G) The total energy of the parcel at the given orbit radius can be calculated by summing the potential energy and the kinetic energy of the parcel.
H) The time it takes for the parcel to revolve about planet P in the given orbit can be found using the formula:
Time = (2 * π * orbit radius) / velocity
Note: Remember to convert the given values to the appropriate units when necessary.
By following these steps, you should be able to find the answers to each question.