A massless pan hangs from a spring that is suspended from the ceiling. When empty, the pan is 50 cm below the ceiling. If a 105 g clay ball is placed gently on the pan, the pan hangs 60 cm below the ceiling. Suppose the clay ball is dropped from the ceiling onto an empty pan. What is the pan's distance from the ceiling when the spring reaches its maximum length?

To determine the pan's distance from the ceiling when the spring reaches its maximum length, we need to understand the concept of equilibrium position for a mass-spring system.

The equilibrium position is the position where the net force on an object is zero. In a mass-spring system, the equilibrium position is reached when the spring is neither stretched nor compressed and the system is at rest.

From the given information, we can observe that when the pan is empty, it hangs 50 cm below the ceiling, which can be considered as the equilibrium position for the system. When a clay ball is placed gently on the pan, the pan hangs 60 cm below the ceiling, indicating that the system is no longer at equilibrium.

Now, let's calculate the spring constant (k) using Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as follows:

F = -kx

Where F is the force, k is the spring constant, and x is the displacement from the equilibrium position in meters.

In this case, we can calculate the spring constant (k) as follows:

F = mg

Where m is the mass (105 g = 0.105 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).

F = (0.105 kg)(9.8 m/s²) = 1.029 N

Since the force (F) exerted by the spring is equal and opposite to the weight of the clay ball, we have:

F = -kx

1.029 N = -k(0.60 m)

Solving for the spring constant (k):

k = 1.029 N / 0.60 m ≈ 1.715 N/m

Now that we have the spring constant (k) for the system, we can determine the pan's distance from the ceiling when the spring reaches its maximum length. At the maximum length, the spring is stretched to its limit, and the force exerted by the spring (kx) is equal to the weight of the clay ball (mg).

So:

kx = mg

1.715 N/m * x = (0.105 kg)(9.8 m/s²)

Solving for x:

x = (0.105 kg)(9.8 m/s²) / 1.715 N/m ≈ 0.601 m

Therefore, the pan's distance from the ceiling when the spring reaches its maximum length is approximately 0.601 meters (or 60.1 cm).