Two uniform 81.6 g marbles 1.81 cm in diameter are stacked in a container that is 2.78 cm wide.
1) Find the force that the container exerts on the marble at the point of contact A, B and C.
2) What force does each marble exert on the other?
Here's the figure:
session [dot] masteringphysics [dot] com/problemAsset/1039128/5/YF-11-55 [dot] jpg
To solve these problems, we need to consider the forces acting on each marble. In this case, we have three points of interest: A, B, and C.
1) To find the force that the container exerts on the marble at each point of contact, we can use the concept of normal force. The normal force is the force exerted by a surface perpendicular to the object in contact with it. In this case, the container exerts a normal force on the marble.
At points A and C:
The force exerted by the container on the marble is equal to the weight of the marble. Therefore, we need to calculate the weight of the marble first.
The weight (W) of an object can be determined using the formula:
W = m * g
Where:
W is the weight,
m is the mass of the object, and
g is the acceleration due to gravity (approximately 9.8 m/s²).
The mass of one marble is 81.6 g, which we need to convert to kilograms:
m = 81.6 g / 1000 = 0.0816 kg
Now we can calculate the weight of the marble at points A and C:
W = m * g = 0.0816 kg * 9.8 m/s² = 0.79968 N (approximately 0.80 N)
Therefore, the force that the container exerts on the marble at points A and C is approximately 0.80 N.
At point B:
At this point, the marble is not in contact with the container. Therefore, the force exerted by the container at this point is zero.
2) To find the force each marble exerts on the other, we can use Newton's third law of motion, which states that every action has an equal and opposite reaction. In this case, the force exerted by one marble on the other will be equal in magnitude but opposite in direction.
The force exerted by one marble on the other is the gravitational force between them. This force can be calculated using the formula for gravitational force:
F = (G * m1 * m2) / r²
Where:
F is the force,
G is the gravitational constant (approximately 6.67430 x 10^-11 m³/(kg s²)),
m1 and m2 are the masses of the marbles, and
r is the distance between the centers of the marbles.
The masses of the marbles are both 81.6 g, which we convert to kilograms:
m1 = m2 = 81.6 g / 1000 = 0.0816 kg
The diameter of each marble is 1.81 cm, so the radius is half of that:
r = 1.81 cm / 2 = 0.905 cm = 0.00905 m
Now we can calculate the force exerted by one marble on the other:
F = (G * m1 * m2) / r² = (6.67430 x 10^-11 m³/(kg s²)) * (0.0816 kg)² / (0.00905 m)²
Using this equation, we can find the force exerted by one marble on the other.