Two isolated masses, M1 = 2.20 kg and M2 = 589 kg are initially rest, a distance d= 161 cm apart. Their gravitational attraction is the only force acting. Calculate the time it takes for M1 to move from that distance to 159 cm from M2. Assume that M2 does not move and that the force is constant over that small distance, and equal to that at 160 cm.
To calculate the time it takes for mass M1 to move from 161 cm to 159 cm from M2, we can use Newton's second law of motion and the law of universal gravitation.
Step 1: Calculate the gravitational force between M1 and M2 at a distance of 161 cm.
The gravitational force (F) between two masses (m1 and m2) is given by the equation:
F = G * (m1 * m2) / r^2
Where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.
In this case, we have:
m1 = 2.20 kg
m2 = 589 kg
r = 161 cm = 1.61 meters
The gravitational constant, G, is approximately 6.674 × 10^-11 N(m/kg)^2.
Plugging in the values, we can calculate the gravitational force between the two masses at a distance of 161 cm.
F = (6.674 × 10^-11 N(m/kg)^2) * ((2.20 kg) * (589 kg)) / (1.61 meters)^2
Step 2: Calculate the acceleration (a) of M1 using Newton's second law of motion.
Newton's second law states that the force acting on an object is equal to the product of its mass and acceleration:
F = m * a
In this case, the force acting on M1 is the gravitational force calculated in step 1.
So, we have:
F = gravitational force between M1 and M2
m = mass of M1 (2.20 kg)
Plugging in the values, we can find the acceleration (a) of M1.
F = (gravitational force between M1 and M2)
m = 2.20 kg
Step 3: Calculate the time (t) using the equation of motion.
To calculate the time it takes for M1 to move from 161 cm to 159 cm, we can use the equation of motion:
d = v * t + (1/2) * a * t^2
Where d is the distance traveled, v is the initial velocity, t is the time, a is the acceleration.
In this case, we want to find the time (t) it takes for M1 to move a distance of 2 cm (161 cm to 159 cm).
d = 2 cm = 0.02 meters
v = 0 (initially at rest)
a = acceleration of M1
Now, we can rearrange the equation to solve for time (t):
t = sqrt((2 * d) / a)
Plugging in the values, we can calculate the time (t) it takes for M1 to move from 161 cm to 159 cm.
t = sqrt((2 * 0.02 meters) / (acceleration of M1))
By following these steps, you can calculate the time it takes for M1 to move from 161 cm to 159 cm from M2.
This is just the same as figuring the time it takes for an object to fall 2cm on earth, except that the gravitation acceleration is not 9.81 m/s^2, but
GM/r^2
where you have the values for G,M and r. Once you have that evaluated,
s = 1/2 at^2
and you can just plug in s=0.02m and a is the acceleration figured above.