A student proposes to measure the gravitational constant G by suspending two spherical objects from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. Draw a free-body diagram of one of the objects. (Do this on paper. Your instructor may ask you to turn in this work.)
If two 130.0 kg objects are suspended at the lower ends of cables 40.00 m long and the cables are attached to the ceiling 1.000 m apart, what is the deflection of each object? (Use 1.000 m for the separation distance in the expression for the gravitational force and assume that the deflection is extremely small compared to 1 m.)
To draw a free-body diagram of one of the objects, we need to consider the forces acting on it.
1. First, draw a circle to represent the spherical object.
2. Draw a downward arrow at the center of the object to represent its weight (mg), which is equal to the gravitational force acting on it.
3. Draw two diagonal arrows on either side of the object, representing the tension in the cables. The arrows should be angled slightly towards the center.
4. Label the downward arrow as "mg" and the diagonal arrows as "T1" and "T2".
The diagram should show the weight of the object downwards and the tension forces pulling towards the center, balancing the weight.
Now, to calculate the deflection of each object, we can use the concept of the gravitational constant G.
The gravitational force between two objects is given by the equation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant,
m1 and m2 are the masses of the objects, and
r is the separation distance between the objects.
In this case, the two objects are identical and have a mass of 130.0 kg each. The separation distance (r) is given as 1.000 m.
The deflection of each object can be determined by calculating the net force acting on it, using the equation:
F_net = T1 + T2 - mg
Since the deflection is extremely small compared to 1 m, we can assume that T1 and T2 are of equal magnitude. Therefore, we can rewrite the equation as:
F_net = 2T - mg
Where:
T is the magnitude of tension in the cables.
To calculate the deflection, we need to solve for T:
T = (F_net + mg) / 2
Substituting the values:
F_net = 0 (since the deflection is small)
m = 130.0 kg
g = 9.8 m/s^2
T = (0 + (130.0 kg * 9.8 m/s^2)) / 2
Plug in these values to calculate the tension in the cables.
After finding the tension (T), you can use trigonometry to determine the deflection of each object. Since the deflection is extremely small, we can assume that the deflection angle is equal to the angle between the cables and the vertical. Use basic trigonometric functions like sine or tangent to calculate the angle and then calculate the deflection.