m1 = 2.7 kg block slides on a frictionless horizontal surface and is connected on one side to a spring (k = 40 N/m) as shown in the figure above. The other side is connected to the block m2 = 3.6 kg that hangs vertically. The system starts from rest with the spring unextended.
a) What is the maximum extension of the spring?
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To find the maximum extension of the spring, we need to analyze the forces acting on the system and apply Newton's laws of motion. Here's how we can approach it step by step:
1. Determine the gravitational force on m2:
The gravitational force on m2 can be calculated using the equation:
F_gravity = m2 * g
where m2 is the mass of the hanging block (3.6 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Calculate the acceleration of the system:
Since the system starts from rest, the net force acting on the system will cause acceleration:
Net Force = F_gravity - Force of the spring (Fs) = (m1 + m2) * a
where m1 is the mass of the sliding block (2.7 kg), m2 is the mass of the hanging block (3.6 kg), and a is the acceleration of the system.
3. Calculate the force exerted by the spring (Fs):
The force exerted by the spring can be found using Hooke's Law:
Fs = k * x
where k is the spring constant (40 N/m) and x is the displacement or extension of the spring.
4. Substitute the values into the equations:
Substituting the known values into the equations, we have:
F_gravity - Fs = (m1 + m2) * a
F_gravity - k * x = (m1 + m2) * a
5. Find the acceleration (a):
Rearrange the equation to solve for the acceleration (a):
a = (F_gravity - k * x) / (m1 + m2)
6. Calculate the maximum extension of the spring (x):
At the maximum extension, the net force acting on the system will be zero. Thus, setting the equation equal to zero and solving for x will give us the maximum extension of the spring:
F_gravity - k * x = 0
x = F_gravity / k
7. Substitute the values and calculate the answer:
Plug in the values for F_gravity and k from step 1 and step 3 respectively and calculate the maximum extension (x) of the spring.
Using the given values, you can follow these steps and plug in the values to find the maximum extension of the spring.