Pendulum Damped by Air Resistance

[cboystills]

Homework Statement

I need to develop an equation for the damping factor with respect to time for a pendulum that consists of a spherical mass attached to a string that is damped by air resistance.

Homework Equations

We are given that the curve fit for the drag coefficient is c_D=0.45+30/Re^0.886, where c_D is the drag coefficient and Re is the Reynold's number.

The equation for the Reynold's number is Re=VD/v, where V is the velocity of the sphere, D is the diameter of the sphere, and v is the kinematic viscosity of air.

We know that the drag force is F_D=(c_DρV²A)/2, where ρ is the density of air, and A=piD²/4.

The Attempt at a Solution

Since the damping force relative to velocity is F_D=-bV, we then have:

b=(c_DρVA)/2.

where b is the damping factor. Substituting our curve fit for the drag coefficient we have:

b=[(0.45+30/Re^0.886)ρVA]/2.

Substituting the Reynold's number formula we have:

b=[(0.45+30/(VD/v)^0.886))ρVA]/2.

I am stuck after this point. I need to find the velocity with respect to time and substitute it into the above equation. However, the Reynold's number changes with respect to velocity. I'm not sure how to proceed. I would appreciate any advice.

 

[anathema]

Dear student,

Thank you for your post. Developing an equation for the damping factor with respect to time for a pendulum with air resistance can be a complex task, but I will guide you through the process.

Firstly, it is important to understand the factors that affect the damping of the pendulum. In this case, the main factor is air resistance, which is dependent on the velocity of the pendulum. As the pendulum swings, its velocity changes, and so does the air resistance. This means that the damping factor will also change with respect to time.

To develop an equation for the damping factor with respect to time, we need to consider the forces acting on the pendulum at any given time. These forces are the gravitational force, the tension force from the string, and the air resistance force. The equation for the damping factor can be derived from the equation of motion for the pendulum, which is:

md2x/dt2 = -mgLsinθ - bv - kx

Where m is the mass of the pendulum, L is the length of the string, g is the acceleration due to gravity, θ is the angle of the pendulum, b is the damping factor, k is the spring constant, and x is the displacement of the pendulum from its equilibrium position.

We can simplify this equation by assuming that the angle of the pendulum is small, which is usually the case for small oscillations. This means that sinθ ≈ θ, and we can rewrite the equation as:

md2x/dt2 = -mgLθ - bv - kx

Now, we need to express the damping force in terms of the velocity of the pendulum. As you correctly stated in your attempt, the drag force is given by FD=(cDρV2A)/2. However, in this case, the velocity of the pendulum is not the same as the velocity of the sphere. The velocity of the pendulum can be expressed as v = ωLθ, where ω is the angular velocity of the pendulum. Substituting this into the drag force equation, we get:

FD=(cDρ(ωLθ)2A)/2

Simplifying this, we get:

FD=(cDρω2L2Aθ2)/2

Since we are assuming small angles, we can further simplify this to:

FD=(cDρω2L2Aθ