Two spheres of equal mass, A and B, are projected off the edge of a 1.0 m bench. Sphere A has a horizontal velocity of 10 m/s
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1) The forces are equal (Newton's third law of motion)
2) The force between the spheres will quadruple
3) The force of gravity exerted by the notebook on you is negligible
Explanation:
1)
In this part of the problem, we want to compare the gravitational force exerted by the larger mass sphere on the smaller mass sphere to the force exerted by the smaller mass sphere to the larger mass sphere.
We can do this by using Newton's third law of motion, which states that:
<em>"When an object A exerts a force (called </em><em>action</em><em>) on an object B, then object B exerts an equal and opposite force (called </em><em>reaction</em><em>) on object A"</em>
In this problem, we can identify the larger mass sphere as object A and the smaller mass sphere as object B. This law tells us that the two forces are equal in magnitude and opposite in direction: therefore, the gravitational force exerted by the larger mass sphere on the smaller mass sphere is equal to the force exerted by the smaller mass sphere to the larger mass sphere.
2)
The magnitude of the gravitational force between the two spheres is given by
where
G is the gravitational constant
are the masses of the two spheres
r is the separation between the two spheres
In this problem, we are asked to find what happens when the distance between the spheres is halved, therefore when the new distance is
Substituting into the equation, we find
So, the force between the two spheres will quadruple.
3)
We can give an estimate for the gravitational force exerted by your notebook on you.
As we said, the magnitude of the gravitational force is
Where:
is the gravitational constant
Let's estimate the following:
is your mass
is the mass of the notebook
, assuming the notebook is at 1 metre from you
Substituting,
We see that this force has an extremely small value: therefore, it is almost negligible in daily life, where other much stronger forces act on you.
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a) The acceleration of gravity is
b) The critical velocity is 6668 m/s (24,006 km/h)
c) The period of the orbit is 8452 s
d) The satellite completes 10.2 orbits per day
e) The escape velocity of the satellite is 9430 m/s
f) The escape velocity of the rocket is 11,191 m/s
Explanation:
a)
The acceleration of gravity for an object near a planet is given by
where
G is the gravitational constant
M is the mass of the planet
R is the radius of the planet
h is the height above the surface
In this problem,
(mass of the Earth)
(Earth's radius)
(altitude of the satellite)
Substituting,
b)
The critical velocity for a satellite orbiting around a planet is given by
where we have again:
(mass of the Earth)
(Earth's radius)
(altitude of the satellite)
Substituting,
Converting into km/h,
c)
The period of the orbit is given by the circumference of the orbit divided by the velocity:
where
v = 6668 m/s
Substituting,
d)
One day consists of:
While the period of the orbit is
T = 8452 s
So, the number of orbits completed by the satellite in one day is
e)
The escape velocity for an object in the gravitational field of a planet is given by
where here we have:
Substituting, we find
f)
We can apply again the formula to find the escape velocity for the rocket:
Where this time we have:
, because the rocket is located at the Earth's surface, so its altitude is zero.
And substituting,
Learn more about gravitational force:
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