1. Two identical 15.0 kg balls, each 25.0 cm in diameter, are suspended by two 35.0 cm wires as shown in the figure. The entire apparatus is supported by a single 18.0 cm wire, and the surfaces of the balls are perfectly smooth.

Find the tension in the left 35.0 cm wire.

2. What is the tension in the right 35.0 cm wire?

3. What is the tension in the top 18.0 cm wire?

4. How hard does each ball push on the other one?

1. The tension in the left 35.0 cm wire is 375 N.

2. The tension in the right 35.0 cm wire is 375 N.
3. The tension in the top 18.0 cm wire is 750 N.
4. Each ball pushes on the other one with a force of 750 N.

To find the tension in the wires and the force exerted by each ball on the other, we need to analyze the forces acting on the system.

1. To find the tension in the left 35.0 cm wire:
- Start by drawing a free-body diagram for the left ball.
- The weight of the left ball acts downward, and the tension in the left wire acts upward.
- Since the ball is in equilibrium, the net force on it is zero.
- The vertical components of the tension and weight should balance each other, so we can set up an equation:
Tension in the left wire = Weight of the left ball
- The weight (W) is given by W = mass * gravity, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Substitute the given mass of the ball into the equation and solve for the tension in the left wire.

2. To find the tension in the right 35.0 cm wire:
- Use the same method as above, but this time consider the right ball.
- Draw a free-body diagram for the right ball.
- The weight of the right ball acts downward, and the tension in the right wire acts upward.
- Set up an equation for equilibrium, similar to the one used in step 1, and solve for the tension in the right wire.

3. To find the tension in the top 18.0 cm wire:
- On the top ball, the combined tension in the two 35.0 cm wires acts in opposite directions to the weight of the ball.
- Consider the forces on the top ball: the weight downward and the two tensions upward.
- Write an equation for equilibrium and solve for the combined tension in the two 35.0 cm wires.
- In this case, the 18.0 cm wire simply provides support and does not contribute to the tension.

4. To find how hard each ball pushes on the other:
- Each ball exerts a force on the other in order to stay in equilibrium.
- The normal force between the balls is equal to the force each ball exerts on the other.
- Since the surfaces of the balls are perfectly smooth, there is no friction between them.
- Therefore, the normal force between the balls is equal to the force they exert on each other.

Note: It's important to consider the angles and forces acting on each ball separately when analyzing the system.

To find the tension in each wire and the force exerted by each ball on the other, we can use the principles of equilibrium and Newton's third law.

1. The tension in the left 35.0 cm wire:
Since the system is in equilibrium, the net force acting on each ball must be zero. The tension in the left wire is equal to the weight of the left ball plus the force exerted by the right ball on the left ball (due to Newton's third law).
Let's calculate the weight of one ball:
Mass (m) = 15.0 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Weight (W) = m * g = 15.0 kg * 9.8 m/s^2 = 147 N

Now we need to find the force exerted by the right ball on the left ball. Since the surfaces of the balls are perfectly smooth, there is no friction between them. This means the only horizontal force between the balls is the force exerted by the right ball on the left ball. This force is equal in magnitude but opposite in direction to the force exerted by the left ball on the right ball.

The force exerted by each ball on the other ball has the same magnitude but opposite direction. To find this force, we'll use the equation:
Force = mass * acceleration

The acceleration is centripetal acceleration due to the circular motion of the balls, which is given by:
Acceleration (a) = v^2 / r

The velocity (v) can be calculated using the equation:
v = 2πr / T
where r is the radius of the ball and T is the time period of one complete revolution (which we'll assume to be the same for both balls).

Radius (r) = 25.0 cm = 0.25 m
Time period (T) = ?
We don't have the value of T in the question, so we'll need to determine it based on the given information in the figure or assume a reasonable value.

Once we have the value of T, we can calculate the velocity (v) and then the acceleration (a). Using this acceleration, we can find the magnitude of the force exerted by each ball on the other. This force will be exerted in the direction opposite to that of the tension in the left wire.

2. The tension in the right 35.0 cm wire:
The tension in the right wire is equal to the weight of the right ball plus the force exerted by the left ball on the right ball. Using the same calculations as in step 1, we can find the weight of the right ball and the force exerted by the left ball on the right ball. The force exerted by the left ball on the right ball will be exerted in the direction opposite to that of the tension in the right wire.

3. The tension in the top 18.0 cm wire:
The tension in the top wire is equal to the sum of the tensions in the left and right wires. This is because the vertical forces must also balance in equilibrium. Since there is no vertical component of force between the balls, the total tension exerted on the top wire is simply the sum of the tensions in the left and right wires.

4. How hard does each ball push on the other one:
The force exerted by each ball on the other is equal in magnitude but opposite in direction, as mentioned earlier. It can be calculated using the equation: Force = mass * acceleration, where the acceleration is the centripetal acceleration as determined in step 1.