Newton’s Law of Gravity specifies the magnitude of the interaction force between two point masses, m1 and m2, separated by the distance r as F(r) = Gm1m2/r^2. The gravitational constant G can be determined by directly measuring the interaction force in the late 18th century by the English scientis Henry Cavendish. This apparatus was a torsion balance consisting of a 6.00-ft wooden rod suspended from a torsion wire, with a lead sphere having a diameter of 2.00 in and a weight of 1.61 lb attached to each end. Two 12.0-in, 348-lb lead balls were located near the smaller balls, about 9.00 in away, and held in place with a separate suspension system. Today’s accepted value for G is 6.674E-11 m^3 kg^-1s^-2.
a) Determine the force of attraction between the larger and smaller balls that had to be measured by this balance.
b) Compare this force to the weight of the small balls.
Ok... so for the first one I have to use the equation given and change the values to kilograms and meters, and use the radii instead of the diameters, this is what I know, but I don't know how to do it when there are three spheres (the big one and the 2 small ones)... I don't get what they are asking for the second part...
Someone please help, it will be deeply appreciated...
To solve this problem, we need to break it down into steps. Let's start with part a.
a) Determine the force of attraction between the larger and smaller balls:
1. Convert the mass of the lead sphere attached to each end of the wooden rod to kilograms:
Mass = 1.61 lb = 1.61 lb * (0.4536 kg / lb) = 0.73 kg
2. Convert the diameter of the lead sphere to meters:
Diameter = 2.00 in = 2.00 in * (0.0254 m / in) = 0.0508 m
3. Calculate the radius of the lead sphere:
Radius = Diameter / 2 = 0.0508 m / 2 = 0.0254 m
4. Calculate the gravitational force using Newton's Law of Gravity:
F(r) = G * m1 * m2 / r^2
where:
G = 6.674E-11 m^3 kg^-1s^-2 (gravitational constant)
m1 = mass of the larger ball = 348 lb = 348 lb * (0.4536 kg / lb) = 157.8 kg (converted to kilograms)
m2 = mass of each smaller ball = 0.73 kg (as calculated earlier)
r = distance between the larger and smaller balls = 9.00 in = 9.00 in * (0.0254 m / in) = 0.2286 m
F(r) = (6.674E-11 m^3 kg^-1s^-2) * (157.8 kg) * (0.73 kg) / (0.2286 m)^2
Calculate the force to find the answer.
Now, let's move on to part b.
b) Compare this force to the weight of the small balls:
1. Calculate the weight of the small balls:
Weight = mass * gravity
where:
mass = 0.73 kg (as calculated earlier)
gravity = 9.8 m/s^2 (acceleration due to gravity)
Weight = 0.73 kg * 9.8 m/s^2
Calculate the weight to find the answer.
I hope this helps! Let me know if you have any further questions.
To solve part a) of the problem, you need to find the force of attraction between the larger and smaller balls using Newton's Law of Gravity:
F(r) = G * m1 * m2 / r^2
Given values:
Mass of the smaller ball, m1 = 1.61 lb
Mass of the larger ball, m2 = 348 lb
Distance between them, r = 9.00 in (convert to meters for consistency)
First, let's convert the masses from pounds to kilograms:
1 lb = 0.4536 kg
m1 = 1.61 lb * 0.4536 kg/lb = 0.731496 kg
m2 = 348 lb * 0.4536 kg/lb = 157.8528 kg
Next, convert the distance between the balls from inches to meters:
1 in = 0.0254 m
r = 9.00 in * 0.0254 m/in = 0.2286 m
Now, plug in the values into Newton's Law of Gravity and solve for the force:
F(r) = G * m1 * m2 / r^2
Given G = 6.674E-11 m^3 kg^(-1) s^(-2), we have:
F(r) = (6.674E-11 m^3 kg^(-1) s^(-2)) * (0.731496 kg) * (157.8528 kg) / (0.2286 m)^2
Calculating this expression will give you the force of attraction between the larger and smaller balls.
For part b), you are asked to compare this force to the weight of the small balls. The weight of an object is given by the mass of the object multiplied by the acceleration due to gravity (g).
The weight of the smaller balls is:
Weight = m1 * g
Assuming g = 9.8 m/s^2 (acceleration due to gravity), you can calculate the weight of the smaller balls by multiplying the mass (m1) by the acceleration due to gravity.
To compare the force of attraction (calculated in part a)) to the weight of the small balls, divide the force by the weight and analyze the result.