A block of mass m = 5 kg is attached to a spring (k = 35 N/m) by a rope that hangs over a pulley of mass M = 7 kg and radius R = 4 cm, as shown in the figure. Treating the pulley as a solid homogeneous disk, neglecting friction at the axle of the pulley, and assuming the system starts from rest with the spring at its natural length, answer the following.
(a) Find the speed of the block after it falls 1 m.
(b) Find the maximum extension of the spring
(a) MgX - (1/2)kX^2 = increase in kinetic energy = (1/2) M V^2 + (1/2) I w^2
Solve for V
M = 5 kg
g = 9.8 m/s^2
X = 1 m
k = 35 N/m
I = (1/2)*Mpulley)*Rpulley^2
w = V/Rpulley
b) Maximum extension Xmax occurs when V = 0
To find the speed of the block after it falls 1 m, we can use the principle of conservation of energy.
(a) Find the speed of the block after it falls 1 m:
Step 1: Calculate the potential energy gained by the block:
The potential energy gained by the block as it falls 1 m is given by:
Potential energy gained = mass * gravity * height
Potential energy gained = 5 kg * 9.8 m/s^2 * 1 m
Potential energy gained = 49 J
Step 2: Calculate the spring potential energy:
Since the spring starts at its natural length, it will be compressed by the force exerted on it due to the block and the pulley. The spring potential energy can be calculated using Hooke's Law:
Spring potential energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its natural length.
Step 3: Calculate the pulley's rotational kinetic energy:
The pulley starts from rest, so its initial rotational kinetic energy is zero. As the block falls, the pulley rotates and gains rotational kinetic energy.
Step 4: Apply conservation of energy:
The total mechanical energy of the system is conserved, and it can be calculated as the sum of potential energy and kinetic energy. At the bottom of the fall, all of the potential energy is converted into kinetic energy (both linear and rotational).
Total mechanical energy = Potential energy gained + Spring potential energy + Pulley's rotational kinetic energy
Total mechanical energy = 49 J + (1/2) * k * x^2 + (1/2) * I * omega^2
where I is the moment of inertia of the pulley and omega is its angular velocity.
Step 5: Solve for the speed of the block:
The linear speed of the block is equal to the linear speed of the pulley's edge. We can relate the linear speed of the block to the angular speed of the pulley using the equation:
v_block = R * omega
where R is the radius of the pulley.
By equating the total mechanical energy to the kinetic energy, we can solve for the speed of the block (v_block):
Total mechanical energy = (1/2) * m * v_block^2
49 J + (1/2) * k * x^2 + (1/2) * I * (v_block / R)^2 = (1/2) * m * v_block^2
Now, let's move on to part (b).
(b) Find the maximum extension of the spring:
When the system reaches its maximum extension, the pulley stops rotating, and the block is at its lowest point. This means that the rotational kinetic energy of the pulley is zero.
Setting the kinetic energy equation equal to zero, we have:
49 J + (1/2) * k * x^2 = (1/2) * m * v_max^2
Solving for the maximum extension (x_max) of the spring, we can use the equations derived in part (a):
49 J + (1/2) * k * x_max^2 = (1/2) * m * v_max^2
We can use this equation to solve for x_max given the values of the mass of the block (m), spring constant (k), and speed of the block (v_max).
To find the speed of the block after it falls 1 m, we need to consider the conservation of mechanical energy.
First, let's find the potential energy of the block when it falls 1 m. Since the block starts from rest with the spring at its natural length, the potential energy stored in the spring is zero initially. Therefore, the total potential energy of the system can be written as:
PE = m * g * h,
where m is the mass of the block, g is the acceleration due to gravity, and h is the height the block falls (which is 1 m in this case).
Next, we can find the kinetic energy of the block when it reaches a speed v. The kinetic energy can be expressed as:
KE = (1/2) * m * v^2.
Since mechanical energy is conserved, the total potential energy at the top of the fall (PE) is equal to the total kinetic energy at the bottom of the fall (KE). Therefore, we can set up the equation as:
m * g * h = (1/2) * m * v^2.
From this equation, we can solve for the speed of the block:
v = sqrt(2 * g * h).
Plug in the values for g (approximately 9.8 m/s^2) and h (1 m) to calculate the speed.
For part (b), 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 its displacement from its equilibrium position.
The force exerted by the spring can be written as:
F = -k * x,
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, the block attached to the spring is pulled downward, causing the spring to stretch. The maximum extension of the spring will occur when the gravitational force pulling the block down is balanced by the force exerted by the spring pulling the block up.
So, we can set up an equation to find the maximum extension of the spring:
m * g = k * x_max.
Rearrange the equation to solve for x_max:
x_max = (m * g) / k.
Plug in the values for m (5 kg), g (approximately 9.8 m/s^2), and k (35 N/m) to calculate the maximum extension of the spring.