A pulley (a soild disk) with mass 2kg and radius 0.3 m is attached by massless rope to two blocks of masses m1=1kg and m2=3 kg. what is the acceleration of block 1 after the system is realeased from rest?
To find the acceleration of block 1, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
In this system, there are two forces acting on block 1: the tension in the rope (T1) and the force of gravity (mg1). The tension in the rope is related to the acceleration of block 1, while the force of gravity opposes the motion.
We can start by calculating the force of gravity acting on block 1:
Force of gravity (mg1) = mass (m1) * acceleration due to gravity (g)
Given that m1 = 1 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can find the force of gravity:
mg1 = 1 kg * 9.8 m/s^2 = 9.8 N
Now, since block 1 is attached to the pulley, the tension T1 in the rope creates a torque on the pulley, causing it to rotate and exert a torque on block 1. The torque can be found by multiplying the tension in the rope by the radius of the pulley:
Torque (τ) = T1 * r
Given that the radius of the pulley is 0.3 m, we can write:
τ = T1 * 0.3 m
Next, we can relate the torque to the angular acceleration of the pulley using the moment of inertia (I) of a solid disk:
τ = I * α
For a solid disk rotating around its central axis, the moment of inertia is given by:
I = (1/2) * m * r^2
Substituting the given values, we get:
τ = (1/2) * 2 kg * (0.3 m)^2 * α
At this point, we can notice that the angular acceleration (α) of the pulley is related to the linear acceleration (a) of block 1 by the following equation:
a = α * r
So, we can rewrite the equation for torque as:
τ = (1/2) * 2 kg * (0.3 m)^2 * (a / r)
Now, we equate the torque from the tension in the rope to the torque exerted by the force of gravity:
T1 * 0.3 m = (1/2) * 2 kg * (0.3 m)^2 * (a / 0.3 m)
Simplifying the equation, we get:
T1 = (1/2) * 2 kg * 0.3 m * a
T1 = 0.3 kg * a
Now, we can sum up the forces acting in the vertical direction (the positive direction is taken upwards):
T1 - mg1 = m1 * a
Substituting the values we have, we get:
0.3 kg * a - 9.8 N = 1 kg * a
Solving for a, we find:
0.3 kg * a - 1 kg * a = 9.8 N
-0.7 kg * a = 9.8 N
Now, dividing both sides of the equation by -0.7 kg, we get:
a = 9.8 N / -0.7 kg
a ≈ -14 m/s²
Therefore, the acceleration of block 1 is approximately -14 m/s².