A spring with 53 hangs vertically next to a ruler. The end of the spring is next to the 21- mark on the ruler. If a 3.0- mass is now attached to the end of the spring, where will the end of the spring line up with the ruler marks?
To determine where the end of the spring will line up with the ruler marks when a 3.0-kg mass is attached, we need to consider the concept of equilibrium in a spring-mass system.
Equilibrium in a spring-mass system occurs when the forces acting on the system are balanced. In this case, the weight of the mass is balanced by the spring force.
First, let's find the spring constant (k) of the spring. The spring constant represents the stiffness of the spring and can be determined by Hooke's Law:
F = -kx
where F is the force applied to the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
From the given information, we know that when the spring is hanging vertically with no mass attached (i.e., at equilibrium), the displacement of the spring is 53 cm from the ruler's end to the spring's end.
Since the spring is in equilibrium, the force applied by the spring (F) is equal in magnitude to the weight of the mass. The weight of the mass can be calculated using the formula:
Weight = mass * acceleration due to gravity
Weight = 3.0 kg * 9.8 m/s^2 (approximately)
Next, we can equate the spring force and the weight of the mass:
-k * 53 cm = 3.0 kg * 9.8 m/s^2
Now, we convert the displacement from centimeters to meters:
-0.53 m * k = 3.0 kg * 9.8 m/s^2
Simplifying the equation:
k = (3.0 kg * 9.8 m/s^2) / 0.53 m
Calculating the value of k:
k ≈ 55.283 m/s^2
Now that we have the spring constant, we can determine the displacement of the spring with the 3.0-kg mass attached. We'll use the same equilibrium equation:
-kx = 3.0 kg * 9.8 m/s^2
Substituting the value of k:
-55.283 m/s^2 * x = 3.0 kg * 9.8 m/s^2
Simplifying the equation:
x ≈ -0.0536 m (or -5.36 cm)
Therefore, the end of the spring will line up with the 15-mark on the ruler (21 cm - 5.36 cm) when a 3.0-kg mass is attached.