A 20-lb disk has a radius of 1ft and can roll without slip on a rough plane that is inclined by an angle è to the horizontal. The disk is attached to the lower end of a linear spring of stiffness 10lb/in. The upper end of the spring is attached to a wall. Determine the natural period of vibration of the disk.

To determine the natural period of vibration of the disk, we need to consider two factors: the gravitational force acting on the disk and the spring force.

First, let's calculate the gravitational force acting on the disk. The weight of an object is given by the formula: weight = mass * gravity. In this case, since we know the weight is 20 lbs, we can convert it to mass using the formula: mass = weight / gravity. The acceleration due to gravity is approximately 32.2 ft/s^2. Thus, the mass of the disk can be calculated as:

mass = 20 lbs / 32.2 ft/s^2.

Next, we need to calculate the gravitational torque acting on the disk when it is inclined at an angle è. The torque is given by the formula: torque = force * distance. In this case, the force is the component of the gravitational force acting along the direction of the incline, which is equal to mg * sin(è). The distance is the radius of the disk, which is 1 ft. Therefore, the gravitational torque can be calculated as:

torque = (mass * gravity * sin(è)) * radius.

Now, let's consider the spring force. The spring force is given by the formula: force = -k * displacement. Here, the stiffness of the spring is given as 10 lb/in. The displacement is the distance the spring is stretched or compressed from its equilibrium position. In the case of simple harmonic motion, the displacement is equal to the amplitude of the motion. Therefore, the spring force can be calculated as:

force = -10 lb/in * amplitude.

Finally, we need to equate the gravitational torque and the spring force to find the natural period of vibration. The torque exerted by the spring is equal to the moment of inertia multiplied by the angular acceleration. In this case, the moment of inertia of a disk is given by the formula: moment of inertia = 0.5 * mass * radius^2. Therefore, we can equate the gravitational torque and the torque exerted by the spring as:

(mass * gravity * sin(è)) * radius = moment of inertia * angular acceleration.

Rearranging the equation, we get:

angular acceleration = (mass * gravity * sin(è)) * radius / moment of inertia.

The angular acceleration is related to the natural period of vibration by the formula: angular acceleration = 2π / period. Thus, we can solve for the period as:

period = 2π / angular acceleration.

Substituting the expression for angular acceleration derived earlier, we can solve for the natural period of vibration of the disk.