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?
m
b) What is the speed of block m2 when the extension is 65 cm?
120
)OMG
a) Well, Max, that's quite a springy situation you've got there! To find the maximum extension of the spring, we need to consider the equilibrium condition. Since there is no friction and the system starts from rest, the gravitational force on m2 must be balanced by the spring force. So, we can write:
m2g = kx_max
where m2 is the mass of the hanging block (3.6 kg), g is the acceleration due to gravity (9.8 m/s^2), k is the spring constant (40 N/m), and x_max is the maximum extension of the spring.
Let's solve for x_max:
x_max = (m2g) / k
Substituting the given values:
x_max = (3.6 kg)(9.8 m/s^2) / 40 N/m
Calculating that out gives us:
x_max = 0.882 m
So, the maximum extension of the spring is approximately 0.882 meters.
b) Now, to find the speed of block m2 when the extension is 65 cm (or 0.65 m), we can use the principle of conservation of mechanical energy. The total mechanical energy in the system is given by the sum of the potential energy and the kinetic energy:
E_initial = E_final
Initially, both blocks are at rest, so the initial kinetic energy is 0. The potential energy is given by:
PE_initial = m2gh
where h is the vertical distance the block m2 has moved. The final kinetic energy is given by:
KE_final = (1/2)m2v^2
where v is the speed of m2 when the extension is 65 cm.
Equating the initial potential energy and the final kinetic energy, we have:
m2gh = (1/2)m2v^2
Substituting the given values:
(3.6 kg)(9.8 m/s^2)(0.65 m) = (1/2)(3.6 kg)v^2
Simplifying that equation, we find:
v^2 = (2)(9.8 m/s^2)(0.65 m) / 3.6 kg
Taking the square root of both sides:
v = sqrt[(2)(9.8 m/s^2)(0.65 m) / 3.6 kg]
And plugging in the numbers into a calculator gives us:
v ≈ 1.022 m/s
So, the speed of block m2 when the extension is 65 cm is approximately 1.022 m/s.
To solve this problem, we can use the principles of conservation of energy and Hooke's Law.
a) To find the maximum extension of the spring, we can use the principle of conservation of energy. At the maximum extension, the potential energy stored in the spring will be equal to the gravitational potential energy of block m2.
The potential energy stored in the spring can be calculated using Hooke's Law:
Potential energy = (1/2) * k * x^2
where k is the spring constant and x is the extension of the spring.
The gravitational potential energy of block m2 can be calculated using the formula:
Potential energy = m * g * h
where m is the mass of block m2, g is the acceleration due to gravity, and h is the height of block m2.
Setting the potential energy of the spring equal to the potential energy of block m2 and solving for x gives us:
(1/2) * k * x^2 = m * g * h
Substituting the given values:
(1/2) * 40 * x^2 = 3.6 * 9.8 * h
Simplifying the equation, we get:
20 * x^2 = 35.28 * h
Since the blocks start from rest, the velocity of the system will be zero at the maximum extension, which means there is no kinetic energy. Therefore, all the initial potential energy is transferred to the spring and gravitational potential energy. At the maximum extension, the gravitational potential energy will be at its maximum, which is when the mass m2 has zero potential energy.
Since the potential energy of block m2 is given by m * g * h, we can calculate h as:
h = 0.65 m (since the extension is given as 65 cm)
Plugging this into the previous equation, we get:
20 * x^2 = 35.28 * 0.65
Simplifying further:
x^2 = (35.28 * 0.65) / 20
x^2 = 1.1392
Taking the square root of both sides, we find:
x = √(1.1392)
x ≈ 1.07 meters
Therefore, the maximum extension of the spring is approximately 1.07 meters.
b) To find the speed of block m2 when the extension is 65 cm, we can again use the principle of conservation of energy.
At that extension, the potential energy of the spring will be equal to the kinetic energy of block m2.
Potential energy of the spring = (1/2) * k * x^2
Kinetic energy of block m2 = (1/2) * m2 * v^2
Setting these two equal and solving for v gives us:
(1/2) * 40 * (0.65^2) = (1/2) * 3.6 * v^2
Simplifying the equation, we find:
8 * (0.42) = 1.8 * v^2
v^2 = (8 * 0.42) / 1.8
v^2 ≈ 1.87
Taking the square root of both sides, we get:
v ≈ √(1.87)
v ≈ 1.37 m/s
Therefore, the speed of block m2 when the extension is 65 cm is approximately 1.37 m/s.
To find the maximum extension of the spring, we can use the conservation of mechanical energy in the system. At the start, the system is at rest and the spring is unextended, so the total mechanical energy is zero.
The total mechanical energy, E, is given by the sum of the potential energy and kinetic energy:
E = PE + KE
Since there is no potential energy at the start (no height change), we have:
E = KE
The kinetic energy of the system is equal to the sum of the kinetic energies of each block:
KE = (1/2)m1v1^2 + (1/2)m2v2^2
At the maximum extension of the spring, the blocks momentarily come to rest. Therefore, the final velocity of both blocks, v1 and v2, is zero. Therefore, we have:
KE = 0
Setting the equation for kinetic energy equal to zero, we can solve for the extension of the spring at maximum deformation.
(1/2)m1v1^2 + (1/2)m2v2^2 = 0
Substituting the given values for m1 and m2, we have:
(1/2)(2.7)(v1^2) + (1/2)(3.6)(0) = 0
Simplifying, we find:
1.35v1^2 = 0
This equation tells us that v1, the velocity of block m1, must be zero for the total mechanical energy to be zero at maximum deformation.
Therefore, the maximum extension of the spring occurs when block m1 comes to rest.
To find the maximum extension, we need to consider the spring force and apply Hooke's Law.
The spring force is given by:
Fs = -kx
Where:
Fs is the spring force
k is the spring constant
x is the extension or compression of the spring
At maximum extension, the spring force and the weight of block m2 are balanced, so:
Fs = m2g
Substituting the values for m2 and g, we have:
-kx = (3.6)(9.8)
Solving for x, the maximum extension of the spring, we find:
x = (3.6)(9.8)/k
Now, let's calculate the maximum extension of the spring.
Substituting the given value for k (40 N/m), we have:
x = (3.6)(9.8)/(40)
Calculating the value, we find:
x ≈ 0.882 m
Therefore, the maximum extension of the spring is approximately 0.882 meters (or 88.2 cm).
Now, let's move on to part b.
To find the speed of block m2 when the extension is 65 cm, we can use the conservation of mechanical energy again.
At this point, the system has some extension in the spring, and the blocks are moving. The total mechanical energy, E, is given by the sum of the potential energy and kinetic energy:
E = PE + KE
The potential energy is given by:
PE = (1/2)kx^2
The kinetic energy is given by:
KE = (1/2)m1v1^2 + (1/2)m2v2^2
At the given extension of 65 cm, the blocks are not at their maximum velocity, so we cannot set the final velocity of both blocks to zero.
We will have to consider the potential energy and kinetic energy at this intermediate point.
Substituting the given values for k, m1, m2, and the extension x (0.65 m), we can calculate the potential energy:
PE = (1/2)(40)(0.65)^2
Calculating this value, we get:
PE ≈ 8.42 J
Since the initial total mechanical energy is zero (at rest with unextended spring), we can set the potential energy equal to the negative of the kinetic energy:
PE = -KE
Substituting the calculated value for potential energy, we have:
8.42 J = -(1/2)(2.7)v1^2 - (1/2)(3.6)v2^2
Simplifying, we find:
-4.21v1^2 - 1.8v2^2 = 8.42
Since the block m2 is initially at rest, its initial velocity is 0. So we only need to solve for v1.
Simplifying the equation further, we have:
-4.21v1^2 = 8.42
Dividing by -4.21, we get:
v1^2 ≈ -8.42/-4.21
Taking the square root, we find:
v1 ≈ √2 ≈ 1.414 m/s
Therefore, the speed of block m2 when the extension of the spring is 65 cm is approximately 1.414 m/s.