A block of mass m = 3 kg is attached to a spring (k = 35 N/m) by a rope that hangs over a pulley of mass M = 6 kg and radius R = 6 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.
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the following what? My gut feel is that you will be dealing with energy, and force in this.
Block
T is tension between block and pulley)
F = m a
m g - T = m a
T = m(g-a)
T - spring force = Torque/R
T - k x = Torque/R
Torque = I alpha (I is in moment of inertia table for M and R)
T -k x = I alpha/R
a = alpha R
T - k x = I a/R^2
m(g-a) - kx = (I/R^2)a
m g = k x + ma +(I^2/R) a
m g = k x +(m+I^2/R^2)d^2x/dt^2
(This is a spring mass system with equilibrium at x = mg/k and 2 pi f=sqrt(k/(m+I^2/R^2)
but onward
split x into Xo + A sinwt
then mg = kXo for steady result
then
x(t) time function = A sin wt
then
a = d^2x/dt^2 = -Aw^2 sin wt = -w^2 x
0 = k x(t) - (m+I^2/R^2)w^2 x(t)
w^2 = k/(m+I^2/R^2)
like we could already see
so we have a vibration at frequency w/2pi about the point mg/k
To answer the question, we need to break it down into smaller parts and apply relevant concepts. Let's examine each part of the question.
1. Find the acceleration of the block.
2. Find the tension in the rope.
3. Find the angular acceleration of the pulley.
4. Find the total mechanical energy of the system.
1. To find the acceleration of the block, we can use Newton's second law. Since the pulley is massless, we can treat the hanging mass as acting on the block with an effective force of M*g (mass of the pulley times acceleration due to gravity).
The system is connected with a rope, so the block and pulley have the same magnitude of acceleration. We can write the equation as follows:
m*a = T - M*g ...(1)
where m is the mass of the block, a is the acceleration, T is the tension in the rope, and g is the acceleration due to gravity.
2. To find the tension in the rope, we can use the fact that the tension is the same on both sides of the pulley. Since one side of the rope is connected to the block and the other to the pulley, we can write the equation as:
T = I * α / (2R) ...(2)
where I is the moment of inertia of the pulley, α is the angular acceleration, and R is the radius of the pulley.
3. To find the angular acceleration of the pulley, we can use the torque equation. The tension in the rope and the radius of the pulley create a torque that accelerates the pulley. The equation can be written as:
I * α = T * R ...(3)
where I is the moment of inertia of the pulley, α is the angular acceleration, T is the tension in the rope, and R is the radius of the pulley.
4. To find the total mechanical energy of the system, we need to consider the potential energy stored in the spring and the rotational kinetic energy of the pulley. The total mechanical energy is the sum of these two energies:
E_total = 0.5 * k * x^2 + 0.5 * I * ω^2 ...(4)
where k is the spring constant, x is the displacement of the block from its equilibrium position, I is the moment of inertia of the pulley, and ω is the angular velocity of the pulley.
By solving these equations simultaneously, we can find the answers to the given questions.