The blocks shown are released from rest with the spring unstretched. The pulley and the
horizontal surface are frictionless. If k = 427.1 N/m and M = 4.5 kg, what is the maximum
extension of the spring?
To find the maximum extension of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Let's break down the problem and find the force exerted by the spring on the system. In this case, there are two masses involved: M (4.5 kg) and m (unknown). The force exerted by the spring can be expressed as:
F = k * x
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement or extension of the spring.
Now, let's analyze the forces acting on the system. There are two forces in play: the force exerted by the spring (F) and the force due to the weight of the masses.
For M (4.5 kg):
The weight force is given by:
F_weight_M = M * g
For m (unknown):
The weight force is given by:
F_weight_m = m * g
Since the pulley and the horizontal surface are frictionless, the net force acting on the system is zero. Therefore, we can set up the following equation:
F - F_weight_M - F_weight_m = 0
Substituting the weight forces:
k * x - M * g - m * g = 0
Simplifying the equation:
k * x = M * g + m * g
Now, we need to find the value of m. Since there is no external force acting on the system, the acceleration of the system is the same for both masses, and we can set up the following equation:
m * a = M * a
Since the system is initially at rest, the acceleration is zero. Therefore, we can conclude that m = M.
Substituting m = M in the equation:
k * x = M * g + M * g
Simplifying further:
k * x = 2 * M * g
Now we have an equation relating the maximum extension of the spring (x) to the known values of the spring constant (k), the mass (M), and the acceleration due to gravity (g).
To find x, we need to know the value of g. On the surface of the Earth, the acceleration due to gravity is approximately 9.8 m/s².
Substituting the known values:
427.1 N/m * x = 2 * 4.5 kg * 9.8 m/s²
Simplifying:
427.1 N/m * x = 88.2 N
Divide both sides by 427.1 N/m:
x = 88.2 N / 427.1 N/m
Evaluating the expression:
x ≈ 0.206 m
Therefore, the maximum extension of the spring is approximately 0.206 meters.