A small bead of mass m is free to slide along a long, thin rod without any friction. The rod rotates in a horizontal plane about a vertical axis passing through its end at a constant rate of f revolutions per second. Show that the displacement of the bead as a function of time is given by r(t)=A1(e^bt) +A2(e^–bt) , where r is measured from the axis of rotation. Find the expression for the constant b. Also, how would you determine the constants A1 and A2?
To solve this problem, we can use Newton's second law of motion. The force acting on the bead is the centripetal force due to its circular motion, which is provided by the tension in the rod. Let's break down the problem into steps:
Step 1: Determine the centripetal force acting on the bead.
The centripetal force is given by Fc = m * (v^2 / r), where m is the mass of the bead, v is its velocity, and r is the distance of the bead from the axis of rotation. Since the bead is sliding without friction along the rod, its velocity can be expressed as v = ω * r, where ω is the angular velocity of the rotating rod in radians per second. Substituting this into the equation, we get Fc = m * (ω^2 * r).
Step 2: Write the equation of motion for the bead.
Using Newton's second law, F = m * a, where F is the net force acting on the bead and a is its acceleration, we can write Fc = m * a. Rearranging the equation, we have a = (ω^2 * r).
Step 3: Solve the differential equation.
To find the displacement of the bead as a function of time, we need to solve the equation a = d^2r/dt^2. Since a = (ω^2 * r), we have d^2r/dt^2 = (ω^2 * r).
This is a second-order linear homogeneous ordinary differential equation, which can be solved by assuming a solution of the form r(t) = e^(bt).
Step 4: Find the expression for the constant b.
Differentiating r(t) twice with respect to time, we have r''(t) = b^2 * e^(bt). Substituting this into the equation, we get b^2 * e^(bt) = ω^2 * e^(bt). Canceling the common factor of e^(bt), we find b^2 = ω^2. Taking the square root of both sides, we get b = ±ω.
Step 5: Determine the constants A1 and A2.
The general solution to the differential equation is r(t) = A1 * e^(bt) + A2 * e^(-bt), where A1 and A2 are arbitrary constants that need to be determined. Since we have shown that b = ±ω, we can rewrite the solution as r(t) = A1 * e^(ωt) + A2 * e^(-ωt).
To determine the constants A1 and A2, we need initial conditions or additional information. For example, we could be given the initial displacement or initial velocity of the bead. With this information, we can solve for A1 and A2 using the equations r(0) = A1 + A2 and v(0) = ω(A1 - A2), where v(0) is the initial velocity of the bead.
In summary, the displacement of the bead as a function of time is given by r(t) = A1 * e^(ωt) + A2 * e^(-ωt), and the constant b is equal to ±ω. The constants A1 and A2 can be determined by using the initial conditions or additional information provided in the problem.