10.1 Three uniform spheres with masses m1 = 6.0 kg, m2 = 2.0 kg ,
and m3 = 4.0 kg are fixed at the positions shown in the diagram.
Assume they are completely isolated and there are no other masses
nearby.
m_1
|
| 3cm
|
m_2-- P
^ |
1cm |2cm
m_3
(a) What is the magnitude of the acceleration on a particle placed at
the origin (Point P)? What is the direction of this acceleration?
b) For a particle at point P, what speed is required for it to
completely escape the gravitational attraction of the three fixed
masses?
(c) How much energy is required to separate the three masses so
that they are very far apart?
To find the answers to the given questions, let's break them down one by one.
(a) What is the magnitude of the acceleration on a particle placed at the origin (Point P)? What is the direction of this acceleration?
To find the net acceleration at point P, we need to consider the gravitational forces acting on it due to the other three masses.
Let's denote the distance from mass m1 to point P as r1, the distance from mass m2 to point P as r2, and the distance from mass m3 to point P as r3.
The gravitational force exerted by a mass on point P can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the universal gravitational constant (approximately 6.674 x 10^-11 N m^2/kg^2), m1 and m2 are the masses, and r is the distance between them.
Using this formula, we can calculate the gravitational forces exerted by each mass on point P as follows:
Force from mass m1: F1 = G * (m1 * m2) / r1^2
Force from mass m2: F2 = G * (m2 * m3) / r2^2
Force from mass m3: F3 = G * (m1 * m3) / r3^2
Now, to find the net acceleration, we need to sum up these individual forces and divide by the total mass at point P:
Net Force = F1 + F2 + F3
Net Acceleration = Net Force / Total Mass
Total Mass = m1 + m2 + m3
Once we have the net acceleration, we can find its magnitude and direction. The magnitude is simply the absolute value of the net acceleration, and the direction can be determined by considering the direction of the net force.
(b) For a particle at point P, what speed is required for it to completely escape the gravitational attraction of the three fixed masses?
To calculate the escape velocity, we need to consider the gravitational potential energy at point P and convert it into kinetic energy.
The gravitational potential energy between two masses can be calculated as:
PE = -G * (m1 * m2) / r,
where PE is the potential energy, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.
Adding up the potential energies for all the mass combinations gives the total gravitational potential energy at point P:
Total PE = PE1 + PE2 + PE3
To completely escape the gravitational attraction, the particle would need enough kinetic energy to overcome the total potential energy. Therefore, the required speed can be calculated using the conservation of mechanical energy:
Total KE = Total PE
KE = 0.5 * m * v^2,
where KE is the kinetic energy, m is the mass, and v is the speed.
(c) How much energy is required to separate the three masses so that they are very far apart?
To find the energy required to separate the three masses, we need to consider the gravitational potential energy. The potential energy is given by the formula:
PE = -G * (m1 * m2) / r,
where PE is the potential energy, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.
We need to calculate the potential energy for all possible mass combinations (m1-m2, m2-m3, m1-m3), sum them up, and take the absolute value to account for the negative sign. This will give us the total potential energy required to separate the masses.
Total PE = |PE1| + |PE2| + |PE3|
Note that "very far apart" implies that the distance r tends towards infinity.
Please make sure to substitute the given values into the formulas to calculate the specific numerical answers to the questions.