Question

An object of mass 178 kg moves in a smooth

straight tunnel of length 2640 km dug through
a chord of a planet of mass 4.48 x 1024 kg and
radius 8.26 x 106 m.
y
F/m
Find the effective force constant of the har-
monic motion.
The value of gravitational
constant is 6.67259 x 10-¹1 Nm²/kg².
Answer in units of N/m.

Answer 1

Answer: If the mass of the object is 178 kg: k = 9.44*10^-5 N/m

Answer for the mass being 82 kg is not complete, because G was not given.

Step-by-step explanation:

If the mass of the object is 178 kg:

The mass of object (m) = 178 kg

Length of tunnel (l) = 2640 km

***The minimum displacement (its amplitude) of object from its mean position,

A = 2640/2 = 1320 km = 1.320 * 10^6 m

Mass of planet (M) = 4.48 * 10^24 kg

Its Radius (R) = 8.26 * 10^6 m

Gravitational constant (G) = 6.67259 * 10^-11 Nm^2/kg^2

Now, restoring force for given hormonic motion will be

Vector of F = -((GMm)/R^3) * x

Comparing with, vector of F = -k*x, we get

k = (GMm)/R^3

k = (((6.67259*10^-11)*(4.48*10^24)*178)/(8.26*10^6)^3)

k = ((5320.99017*10^(-11+24))/(563.559976*10^18))

k = 9.44*10^(13-18)

k = 9.44*10^-5 N/m

If the mass of the object is 82 kg:

The mass of object (m) = 82 kg

Length of tunnel (l) = 2430 km

***The minimum displacement (its amplitude) of object from its mean position,

A = 2430/2 = 1215 km = 1.215*10^6 m

Mass of planet (M) = 4.16 * 10^24 kg

Its Radius (R) = 7 * 10^6 m

Gravitational constant (G) not given in the photo

Now, restoring force for given hormonic motion will be

Vector of F = -((GMm)/R^3) * x

Comparing with, vector of F = -k*x, we get

k = (GMm)/R^3

k = ((G*(4.16*10^24)*82)/(7*10^6)^3)

Remember that when multiplying two values with exponents, you add the exponents together!

k = ((3.4112*10^26 * G)/(3.43*10^20))

Insert G to solve!

Answer 2

To find the effective force constant of the harmonic motion, calculate the gravitational force acting on the object and equate it to the restoring force of the harmonic motion. The gravitational force is approximately 77.9 N and the effective force constant is approximately 0.0295 N/m.

Step-by-step explanation:

To find the effective force constant of the harmonic motion, we need to calculate the gravitational force acting on the object and equate it to the restoring force of the harmonic motion.

The formula for the gravitational force is given by:

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

Where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

In this case, the two objects are the planet and the object in the tunnel. The distance between them is the radius of the planet plus the length of the tunnel:

r = R + L

Substituting the values, we get:

F = (6.67259 x 10^-11 Nm²/kg²) * ((4.48 x 10^24 kg) * (178 kg)) / ((8.26 x 10^6 m) + (2.64 x 10^6 m))²

Simplifying, we find that the gravitational force is approximately 77.9 N.

The restoring force of the harmonic motion is given by Hooke's Law:

F = -k * x

Where F is the force, k is the force constant, and x is the displacement from equilibrium.

Since the object is in equilibrium, the gravitational force must equal the restoring force:

77.9 N = k * (2L)

Where L is the length of the tunnel.

Simplifying, we find that the effective force constant of the harmonic motion is approximately 0.0295 N/m.