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
anybody? please?
To find the effective force constant of the harmonic motion, we need to use the formula for the force of gravity on an object:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity,
G is the gravitational constant,
m1 is the mass of the first object,
m2 is the mass of the second object, and
r is the distance between the centers of the two objects.
In this case, the object moving in the tunnel is our first object, with a mass of 154 kg, and the planet is our second object, with a mass of 3.19 × 10^24 kg.
First, we need to find the distance between the centers of the two objects. Given that the tunnel is dug through a chord of the planet, we can calculate the chord length using the formula:
chord length = 2 * sqrt(R^2 - r^2)
Where:
R is the radius of the planet, and
r is the radius of the tunnel.
In this case, the radius of the planet is 1 × 10^7 m, and the length of the tunnel is 2990 km, or 2990000 m. Since the tunnel is dug through a chord, the radius of the tunnel is equal to half the length of the tunnel.
Substituting these values into the formula, we get:
chord length = 2 * sqrt((1 × 10^7)^2 - (2990000/2)^2)
Now that we have the chord length, we can find the effective force constant by dividing the force of gravity by the displacement of the object in the tunnel.
To calculate the displacement, we can subtract the length of the tunnel from the chord length:
displacement = chord length - length of the tunnel
Finally, we can calculate the effective force constant:
effective force constant = F / displacement
Substituting the values and calculating the equation will give us the answer in units of N/m.