A 145-kg object and a 445-kg object are separated by 3.90 m.
(a) Find the magnitude of the net gravitational force exerted by these objects on a 34.0-kg object placed midway between them. (N)
(b) At what position (other than an infinitely remote one) can the 34.0-kg object be placed so as to experience a net force of zero from the other two objects?
(m from the 445 kg mass toward the 145 kg mass)
a) net force=G34(445 -145 )/d^2 where d is half the separation distance.
b. let x be the distance from the larger object.
Then the force is zero at that point. 0=G*34(445/x^2-145/(3.9-x)^2
or 445/x^2=145/(3.9-x)^2
(3.9-x)^2=.326 x^2
take sqrt of both sides
(3.9-x)=.571 x
1.571x=3.9
x=2.48m from the larger object
check all that math.
To solve this problem, we'll use Newton's law of universal gravitation:
The gravitational force between two objects can be calculated using the formula:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects
- G is the gravitational constant (approximately equal to 6.674 × 10^-11 N*m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
(a) To find the magnitude of the net gravitational force exerted by the 145-kg and 445-kg objects on a 34.0-kg object placed midway between them, we can calculate the gravitational force between the 34.0-kg object and each of the other objects separately and then sum them up.
First, let's calculate the force exerted by the 445-kg object on the 34.0-kg object. The distance between them is 3.90 m. So, we have:
F1 = (G * m1 * m2) / r1^2
F1 = (6.674 × 10^-11 N*m^2/kg^2) * (34.0 kg) * (445.0 kg) / (3.90 m)^2
Calculate F1 using the above expression.
Second, let's calculate the force exerted by the 145-kg object on the 34.0-kg object. The distance between them is also 3.90 m. So, we have:
F2 = (G * m1 * m2) / r2^2
F2 = (6.674 × 10^-11 N*m^2/kg^2) * (34.0 kg) * (145.0 kg) / (3.90 m)^2
Calculate F2 using the above expression.
Finally, to find the net gravitational force, we sum up these two forces:
F_net = F1 + F2
Calculate F_net using the above expression.
The magnitude of the net gravitational force exerted by these objects on the 34.0-kg object placed midway between them is equal to F_net.
(b) To find the position where the 34.0-kg object experiences a net force of zero from the other two objects, we need to find a point where the gravitational forces from the 145-kg and 445-kg objects cancel out.
Let's assume the distance from the 445-kg mass to the 34.0-kg mass is "x" (m) and the distance from the 34.0-kg mass to the 145-kg mass is "3.90 - x" (m). At this position, the gravitational force from the 445-kg mass should be equal and opposite to the force from the 145-kg mass.
So, we can set up an equation:
F1 = F2
Using the formula for the gravitational force, we have:
(G * m1 * m2) / r1^2 = (G * m1 * m2) / r2^2
Simplifying the equation and solving for "x" will give us the position where the net force on the 34.0-kg object is zero.