A block of mass 0.74 kg is suspended by a string which is wrapped so that it is at a radius of 0.060 m from the center of a pulley. The moment of inertia of the pulley is 0.0061 kg·m2. There is friction as the pulley turns. The block starts from rest, and its speed after it has traveled downwards a distance of D= 0.69 m, is 1.129 m/s. Calculate the amount of energy dissipated up to that point.
Thank's!!
To calculate the amount of energy dissipated, we need to first calculate the work done against friction. The work done against friction is equal to the force of friction multiplied by the distance traveled.
The force of friction can be determined using the equation:
frictional force = coefficient of friction * normal force.
Since the block is being pulled downward by gravity, the normal force is equal to the weight of the block, which is given by:
normal force = mass * gravity.
The coefficient of friction is not given directly, so we need to use the information about the moment of inertia of the pulley to determine it.
The rotational kinetic energy of the pulley is given by:
kinetic energy = (1/2) * moment of inertia * angular velocity^2.
The angular velocity can be determined using the equation:
angular velocity = linear velocity / radius of the pulley.
Since we know the velocity of the block after it has traveled a distance of 0.69 m, we can determine the linear velocity.
The linear velocity is given by:
linear velocity = square root(2 * acceleration * distance),
where the acceleration can be determined from Newton's second law:
acceleration = force / mass.
We already have the mass of the block, so we need to determine the force. The force is given by:
force = mass * acceleration = mass * (gravity - frictional force).
We can substitute this force value back into the equation for acceleration to get the final expression for the linear velocity.
Once we have the linear velocity, we can calculate the angular velocity using the given radius of the pulley.
Now that we have the angular velocity, we can calculate the rotational kinetic energy of the pulley.
Finally, the work done against friction is equal to the change in rotational kinetic energy. Since the pulley starts from rest, the initial rotational kinetic energy is zero.
Therefore, the amount of energy dissipated is equal to the final rotational kinetic energy of the pulley.
Let's calculate these steps one by one.