A 5.9-kg rock and a 3.0 10-4-kg pebble are held near the surface of the earth.
(a) Determine the magnitude of the gravitational force exerted on each by the earth.(b) Calculate the magnitude of the acceleration of each object when released.
(a) Well, let's not rock the boat and calculate that gravitational force. The magnitude of the gravitational force exerted on an object near the surface of the Earth can be found using the formula F = mg, where m is the mass of the object and g is the acceleration due to gravity.
For the rock:
F = (5.9 kg)*(9.8 m/s^2)
F = 57.82 N
For the pebble:
F = (3.0 x 10^-4 kg)*(9.8 m/s^2)
F = 2.94 x 10^-3 N
So, the rock is feeling a force of approximately 57.82 N, while the pebble is experiencing a force of around 2.94 x 10^-3 N.
(b) Now, let's see how fast these objects will accelerate. The magnitude of the acceleration of an object due to gravity can be found using the formula a = F/m, where F is the gravitational force and m is the mass of the object.
For the rock:
a = (57.82 N)/(5.9 kg)
a ≈ 9.81 m/s^2
For the pebble:
a = (2.94 x 10^-3 N)/(3.0 x 10^-4 kg)
a ≈ 9.8 m/s^2
Surprise, surprise! Both the rock and the pebble will have the same magnitude of acceleration, approximately 9.8 m/s^2. Looks like gravity plays no favorites when it comes to falling objects.
To determine the magnitude of the gravitational force exerted on each object by the earth, 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 (6.67430 × 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) For the 5.9 kg rock:
m1 = 5.9 kg
m2 = mass of the earth ≈ 5.97 × 10^24 kg
r = radius of the earth ≈ 6.37 × 10^6 m
Plugging the values into the formula:
F1 = G * (m1 * m2) / r^2
= (6.67430 × 10^-11 N m^2 / kg^2) * (5.9 kg) * (5.97 × 10^24 kg) / (6.37 × 10^6 m)^2
Calculating this:
F1 ≈ 5.89 × 10^2 N (Newtons)
(b) For the 3.0 × 10^-4 kg pebble:
m1 = 3.0 × 10^-4 kg
Using the same mass and radius as above, we can calculate the gravitational force on the pebble using the formula:
F2 = G * (m1 * m2) / r^2
= (6.67430 × 10^-11 N m^2 / kg^2) * (3.0 × 10^-4 kg) * (5.97 × 10^24 kg) / (6.37 × 10^6 m)^2
Calculating this:
F2 ≈ 2.89 × 10^-2 N (Newtons)
Now, to calculate the magnitude of the acceleration of each object when released, we can use Newton's second law of motion:
F = m * a
where F is the net force acting on an object, m is the mass of the object, and a is its acceleration.
(a) For the 5.9 kg rock:
Using Newton's second law and rearranging for acceleration:
a1 = F1 / m1
= (5.89 × 10^2 N) / (5.9 kg)
Calculating this:
a1 ≈ 1.00 × 10^2 m/s^2 (meters per second squared)
(b) For the 3.0 × 10^-4 kg pebble:
Using Newton's second law and rearranging for acceleration:
a2 = F2 / m2
= (2.89 × 10^-2 N) / (3.0 × 10^-4 kg)
Calculating this:
a2 ≈ 9.63 × 10^-2 m/s^2 (meters per second squared)
Therefore:
(a) The magnitude of the gravitational force exerted on the rock by the earth is approximately 589 N.
(b) The magnitude of the acceleration of the rock when released is approximately 100 m/s^2.
(a) The magnitude of the gravitational force exerted on the pebble by the earth is approximately 0.029 N.
(b) The magnitude of the acceleration of the pebble when released is approximately 0.0963 m/s^2.
To determine the magnitude of the gravitational force exerted on each object by the Earth, we can use the equation:
F = m * g
Where:
- F is the gravitational force
- m is the mass of the object
- g is the acceleration due to gravity
(a) Magnitude of the gravitational force exerted on each by the Earth:
For the rock:
- Mass (m1) = 5.9 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
F1 = m1 * g
= 5.9 kg * 9.8 m/s^2
≈ 57.82 N
For the pebble:
- Mass (m2) = 3.0 * 10^-4 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
F2 = m2 * g
= (3.0 * 10^-4) kg * 9.8 m/s^2
≈ 0.00294 N
Thus, the magnitude of the gravitational force exerted on the rock is approximately 57.82 N, and the magnitude of the gravitational force exerted on the pebble is approximately 0.00294 N.
(b) To calculate the magnitude of the acceleration of each object when released, we need to use Newton's second law of motion:
F = m * a
Where:
- F is the net force acting on the object
- m is the mass of the object
- a is the acceleration of the object
In this case, the net force is the gravitational force.
For the rock:
- Gravitational force (F1) = 57.82 N
- Mass (m1) = 5.9 kg
F1 = m1 * a1
57.82 N = 5.9 kg * a1
a1 = 57.82 N / 5.9 kg
a1 ≈ 9.80 m/s^2
For the pebble:
- Gravitational force (F2) = 0.00294 N
- Mass (m2) = 3.0 * 10^-4 kg
F2 = m2 * a2
0.00294 N = (3.0 * 10^-4) kg * a2
a2 = 0.00294 N / (3.0 * 10^-4) kg
a2 ≈ 9.80 m/s^2
Thus, the magnitude of the acceleration of each object when released is approximately 9.80 m/s^2.
(a) F =G•m •M/R²
the gravitational constant G =6.67•10^-11 N•m²/kg²,
Earth’s mass is M = 5.97•10^24 kg,
Earth’s radius is R = 6.378•10^6 m.
(b) g= G•M/R²