An object of mass 154 kg moves in a smooth straight tunnel of length 2990 km dug through a chord of a planet of mass 3.19 × 10^24 kg and radius 1 × 10^7 m
Find the effective force constant of the harmonic motion. The value of the universal gravitational constant is 6.67259 × 10^−11 N · m^2/kg^2.
Answer in units of N/m
To find the effective force constant of the harmonic motion, we need to consider the gravitational force acting on the object in the tunnel.
The gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / R^2
Where:
F is the gravitational force
G is the universal gravitational constant (6.67259 × 10^-11 N · m^2/kg^2)
m1 and m2 are the masses of the objects
R is the distance between the centers of the two objects
In this case, the object inside the tunnel is attracted towards the planet. We can calculate the force using the above formula, with the mass of the object in the tunnel (m1) and the mass of the planet (m2), and the distance between their centers (R).
Let's calculate the force:
F = (G * m1 * m2) / R^2
= (6.67259 × 10^-11 N · m^2/kg^2) * (154 kg) * (3.19 × 10^24 kg) / (1 × 10^7 m)^2
Calculating this expression will give us the gravitational force acting on the object. However, to find the effective force constant of the harmonic motion, we need to relate this gravitational force to the characteristics of the harmonic motion.
In harmonic motion, the force acting on the object is given by Hooke's Law:
F = -k * x
Where:
F is the force
k is the force constant or spring constant
x is the displacement from the equilibrium position
In this case, since the object is moving in a smooth straight tunnel, the restoring force is provided by gravity. Therefore, the gravitational force acting on the object (calculated above) is equal to the restoring force:
F = -k * x
Now, we need to find the displacement (x) of the object in the tunnel. The length of the tunnel is 2990 km, but we also need to take into account the radius of the planet (1 × 10^7 m). The net displacement is the difference between the length of the tunnel and the circumference of the planet:
x = (2990 km - 2 * π * 1 × 10^7 m)
Now, we can calculate the effective force constant (k) by rearranging the formula:
k = -F / x
= -((6.67259 × 10^-11 N · m^2/kg^2) * (154 kg) * (3.19 × 10^24 kg) / (1 × 10^7 m)^2) / [(2990 km - 2 * π * 1 × 10^7 m)]
Calculating this expression will give us the effective force constant of the harmonic motion, which represents the restoring force provided by gravity in the given system.
Please note that the answer will be in units of N/m, as requested.