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 consider the conservation of mechanical energy in the system.
First, let's determine the potential energy of the hanging mass (m2). The potential energy of an object with mass "m" and height "h" is given by the formula PE = m * g * h, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
PE(m2) = m2 * g * h
Since the system starts from rest, the initial energy of the system is zero. Therefore, the total mechanical energy of the system remains constant throughout the motion.
At maximum extension, the block m2 will be at its highest position. At this point, all the energy will be converted into potential energy of the hanging mass.
Total Energy = PE(m2)
Equating the total energy to the potential energy, we can find the maximum extension of the spring.
Total Energy = PE(m2)
0.5 * k * x^2 = m2 * g * h
Where:
k = spring constant (40 N/m)
x = maximum extension of the spring (what we want to find)
m2 = mass of the hanging block (3.6 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = maximum height of the hanging block (what we want to find)
Rearrange the equation to solve for x:
0.5 * k * x^2 = m2 * g * h
x^2 = (2 * m2 * g * h) / k
x = sqrt((2 * m2 * g * h) / k)
Now, we need to find the maximum height (h) of the hanging block. At maximum extension, all the potential energy gets converted into the potential energy of the hanging block.
Total Energy = PE(m2)
0.5 * k * x^2 = m2 * g * h
h = (0.5 * k * x^2) / (m2 * g)
Substituting this value of h in the previous equation, we get:
x = sqrt((2 * m2 * g * ((0.5 * k * x^2) / (m2 * g))) / k)
Simplifying the equation further:
x = sqrt(x^2 / k)
Squaring both sides of the equation:
x^2 = x^2 / k
1 = 1 / k
Therefore, the maximum extension of the spring is 1 meter.