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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b) What is the speed of block m2 when the extension is 65 cm?
To find the maximum extension of the spring, we need to use the conservation of mechanical energy principle.
a) To do this, we first need to determine the total mechanical energy of the system. The system starts from rest, so the initial kinetic energy is zero. The only form of energy in the system is the potential energy of the hanging mass m2.
The potential energy of the hanging mass is given by the equation:
U = m2 * g * h
where m2 is the mass of m2, g is the acceleration due to gravity, and h is the vertical distance the mass m2 has fallen.
In this case, since the system starts from rest, the initial potential energy is equal to the maximum potential energy. Therefore, we can rewrite the equation as:
U = (m1 + m2) * g * h
where m1 is the mass of m1.
The maximum extension of the spring occurs when all the potential energy of m2 is converted into the spring potential energy. This can be expressed as:
U = (1/2) * k * x^2
where k is the spring constant and x is the maximum extension of the spring.
Setting the two equations equal to each other:
(m1 + m2) * g * h = (1/2) * k * x^2
Solving for x, we have:
x = sqrt((2 * (m1 + m2) * g * h) / k)
Substituting the given values, we can calculate the maximum extension of the spring.
b) To find the speed of block m2 when the extension is 65 cm, we need to use the principle of conservation of mechanical energy.
The mechanical energy of the system is given by:
E = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 + (1/2) * k * x^2
where v1 is the velocity of m1, v2 is the velocity of m2, and x is the extension of the spring.
Since the system starts from rest, the initial kinetic energies are both zero. Therefore, the equation simplifies to:
E = (1/2) * m2 * v2^2 + (1/2) * k * x^2
The final mechanical energy is equal to the potential energy of m2 and the potential energy stored in the spring:
E = (1/2) * m2 * v2^2 + m2 * g * h + (1/2) * k * x^2
Setting the initial mechanical energy to the final mechanical energy:
0 = (1/2) * m2 * v2^2 + m2 * g * h + (1/2) * k * x^2
This equation can be rearranged to solve for v2:
v2 = sqrt(-2 * g * h - k * x^2 / m2)
Substituting the given values, we can calculate the speed of block m2.