A block is on the end of a massless cable that is tied to a frictionless wheel (uniform disk) as shown in the figure at right. The wheel has a radius of 0.4 m. The system is initially at rest. If the block moves downward, with a constant acceleration, a distance of 2 m in 3 s, what is the mass of the wheel?
To find the mass of the wheel, we can use Newton's second law of motion.
First, let's understand the forces acting on the system. The block is moving downward, so there must be a force pulling it downwards. This force is provided by the tension in the cable.
The tension in the cable is equal to the force required to accelerate the block. Using Newton's second law, we have:
Tension = mass of the block * acceleration
The block moves downward with a constant acceleration, so we can calculate its acceleration using the kinematic equation:
Distance = (1/2) * acceleration * time^2
Plugging in the given values, we have:
2 m = (1/2) * acceleration * (3 s)^2
Simplifying, we find:
2 = 4.5 * acceleration
Now, we can substitute the acceleration value into the tension equation:
Tension = mass of the block * (4.5 * acceleration)
Using the given information, we also know that the radius of the wheel is 0.4 m. Since the wheel is a uniform disk, we can calculate its moment of inertia using the formula:
Moment of inertia = (1/2) * mass of the wheel * radius^2
The moment of inertia represents an object's resistance to changes in its rotational motion. In this case, we assume that the wheel is initially at rest, so the tension in the cable causes it to start rotating.
The torque applied to the wheel by the tension is given by:
Torque = tension * radius
Now, we can set the torque equal to the change in angular momentum of the wheel to solve for the mass:
Torque = moment of inertia * (change in angular velocity / change in time)
Since the system is initially at rest, the change in angular velocity is equal to the final angular velocity. We can express it as:
Change in angular velocity = (2π radians) / (time taken to rotate completely)
The time taken to rotate completely can be calculated using the linear velocity of the block and the circumference of the wheel:
Time taken to rotate completely = distance / (circumference of the wheel)
Finally, we can substitute the torque and change in angular velocity values into the equation:
Tension * radius = moment of inertia * (change in angular velocity / change in time)
Simplifying, we find:
mass of the wheel = (Tension * radius * (change in time)) / ((2π radians) / (time taken to rotate completely))