a. 2 objects each of mass 200 kg and 500 kg are separated by 50 cm. Where body of mass 20 kg should be placed so that the resultant force of gravity experienced by objects equal to zero?
b. launched a rocket leaving Earth. Known radius of the earth 6.4 x 10 ³ km and acceleration of gravity is 10 m / s ^ 2. big footed rocket set off in order to escape the force of gravity?
c. a body of mass 10 kg below the height of 130 km above the earth's surface. if the radius of the earth 6370 miles, how heavy objects at that height?
d. if the gravitational field at the Earth's surface 9.8 m / s ^ 2. the magnitude of the gravitational field at the height of the surface of the earth is R? (R is the radius of the earth)
a. To find the position where the resultant force of gravity is zero between the two objects, we can use the concept of gravitational force and the principle of superposition. The gravitational force between two objects is given by the equation:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity,
G is the gravitational constant (approximately 6.67 x 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the two objects.
In this case, we have two objects of mass 200 kg and 500 kg, separated by 50 cm (which is equal to 0.5 m).
Let's assume the mass of the third object to be placed is M, and we need to find its position.
To have the resultant force of gravity experienced by the objects equal to zero, the net gravitational force on each object must be equal and opposite. Mathematically, we can write:
F1 = F3 (gravitational force between object 1 and object 3)
F2 = F3 (gravitational force between object 2 and object 3)
Using the equation for gravitational force mentioned above, we can write:
(G * (m1 * M) / r1^2) = (G * (m2 * M) / r2^2)
Substituting the given values, we have:
(6.67 x 10^-11 N m^2/kg^2 * (200 kg * M) / (0.5 m)^2 = (6.67 x 10^-11 N m^2/kg^2 * (500 kg * M) / (0.5 m)^2
Simplifying further:
(200 * M) / (0.5)^2 = (500 * M) / (0.5)^2
Dividing both sides of the equation by (0.5)^2, we get:
200 * M = 500 * M
Simplifying, we find M cancels out:
200 = 500
This is not possible, which means there is no position where the resultant force of gravity experienced by the objects can be made equal to zero.
a. To find the position where a body of mass 20 kg needs to be placed so that the resultant force of gravity experienced by the objects is zero, we need to consider the gravitational forces between the objects and the 20 kg body.
Let's denote the mass of the first object as M1 (200 kg), the mass of the second object as M2 (500 kg), and the mass of the 20 kg body as M. The distance between the objects is given as 50 cm (or 0.5 m).
The gravitational force between two objects is given by the equation F = G * (M1 * M2) / r^2, where G is the gravitational constant and r is the distance between the objects.
Since the resultant force of gravity experienced by the objects should be zero, the sum of the gravitational forces acting on each object should cancel out. Therefore, we can set up the equation F1 + F2 = 0, where F1 is the gravitational force between the first object and the 20 kg body, and F2 is the gravitational force between the second object and the 20 kg body.
F1 = G * (M1 * M) / (0.5)^2
F2 = G * (M2 * M) / (0.5)^2
Setting F1 + F2 = 0, we can solve for the mass M.
b. To calculate the minimum speed required for a rocket to escape the force of gravity, we can use the concept of gravitational potential energy and kinetic energy.
The potential energy at the surface of the Earth is given by the equation PE = m * g * h, where m is the mass, g is the acceleration due to gravity (10 m/s^2), and h is the height from the surface of the Earth.
The kinetic energy is given by the equation KE = (1/2) * m * v^2, where m is the mass and v is the velocity.
To escape the force of gravity, the kinetic energy should be equal to or greater than the potential energy. Therefore, we can set up the equation KE ≥ PE and solve for the minimum velocity v.
c. To calculate the weight of an object at a height of 130 km above the Earth's surface, we can use the concept of gravitational force.
The weight of an object is given by the equation W = m * g, where m is the mass and g is the acceleration due to gravity.
The acceleration due to gravity varies with the distance from the center of the Earth. We can use the concept of gravitational field strength to find the value of g at the given height.
The gravitational field strength is given by the equation g = G * (M / r^2), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth.
By substituting the given values, we can calculate the gravitational field strength at the height of 130 km. Then, we can calculate the weight of the 10 kg object using the equation W = m * g.
d. The magnitude of the gravitational field at the height of the surface of the Earth (radius R) can be calculated using the concept of gravitational field strength.
The gravitational field strength is given by the equation g = G * (M / r^2), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth.
By substituting the given values, we can calculate the gravitational field strength at the height of the surface of the Earth (R).