7. A uniform flexible chain of length l, with weight per unit length lamba, passes over a small, frictionless, massless pulley. It is released from a rest position with a length of chain x hanging from one side and a length l-x from the other side. Find the acceleration a as a function of x.
To find the acceleration a as a function of x, we can start by considering the forces acting on the system.
First, let's analyze the tension in the chain. On the side with length x hanging, the tension will be greater at the top and gradually decrease as we move downwards. Similarly, on the other side with length l-x hanging, the tension will be greater at the top and gradually decrease as we move upwards.
Now, consider an infinitesimally small section of the chain of length dx hanging from the side with length x. The weight of this section is given by lamba * dx * g, where g is the acceleration due to gravity. The tension force acting on the section can be divided into two components: one component pulling upwards due to the tension on the left side of the section, and one component pulling downwards due to the tension on the right side of the section.
Using Newton's second law, we can write the equation of motion for this infinitesimally small section of the chain:
lamba * dx * g - T(x) + T(x + dx) = m * a
Here, m is the mass of the section, and a is the acceleration of the section.
Since the chain is considered to be uniform, we can assume that m is the same for all sections. Also, since dx is infinitesimally small, we can approximate T(x + dx) to be T(x) + dT, where dT is the change in tension over the distance dx.
Substituting these approximations into the equation of motion, we get:
lamba * dx * g - T(x) + T(x) + dT = m * a
Simplifying the equation, we have:
lamba * dx * g + dT = m * a
Now, we need to find the expression for dT.
For a small section of length dx, the slope of the chain at that point is given by dy/dx, where y is the height of the chain. Using geometry, we can determine that dy = dx * d(x/l), which equals dx * (dl - dx)/l.
Since lambda is the weight per unit length of the chain, the weight of this small section is given by lambda * dx * g.
Now, using trigonometry, we can determine that dT/dx = dy * lambda * g * sin(theta), where theta is the angle the chain makes with the vertical direction.
Substituting dy = dx * (dl - dx)/l into the equation for dT/dx, we get:
dT/dx = (dx * (dl - dx)/l) * lambda * g * sin(theta)
Now, we need to find sin(theta). For that, we can consider a right-angled triangle with sides dx and dy. The hypotenuse of this triangle is the arc length ds, which is given by ds = l * d(theta). By using similar triangles, we can find that dy = ds * dx / l.
Substituting this value of dy into the equation for dT/dx, we have:
dT/dx = (dx * (dl - dx)/l) * lambda * g * sin(theta)
= (dx * (dl - dx)/l) * lambda * g * ds * dx / l
Simplifying the equation, we get:
dT = (lambda * g * ds * dx^2 * (dl - dx))/l^2
Now, substituting dT into the equation of motion, we have:
lamba * dx * g + (lambda * g * ds * dx^2 * (dl - dx))/l^2 = m * a
Simplifying the equation, we get:
dx / l + (ds * dx^2 * (dl - dx))/l^2 = m * a / (lambda * g)
Now, we need to find expressions for ds and dl.
For a small section of length dx, ds is given by ds = l * d(theta). Since dy = ds * dx / l, we can determine that dy = (dx * d(theta)) / l. By using similar triangles, we can find that dl = (dx * d(theta)) / l.
Now, substituting ds = l * d(theta) and dl = (dx * d(theta)) / l into the equation, we have:
dx / l + (l * d(theta) * dx^2 * ((dx * d(theta)) / l - dx))/l^2 = m * a / (lambda * g)
Simplifying the equation, we get:
dx + (d(theta) * dx^2 * ((dx * d(theta)) - l * dx))/l = m * a / (lambda * g)
Now, we need to find expressions for d(theta).
For a small section of length dx, we know that d(theta) is given by d(theta) = dx / r, where r is the radius of the pulley.
Now, substituting d(theta) = dx / r into the equation, we have:
dx + ((dx / r) * dx^2 * ((dx * (dx / r)) - l * dx))/l = m * a / (lambda * g)
Simplifying the equation, we get:
dx + (dx^3 * (dx^2/r - l))/ (l * r) = m * a / (lambda * g)
Now, we have an equation relating the length of the chain hanging on one side (x) and the acceleration (a).
Please note that the final equation can be further simplified by considering appropriate values for the length of the chain (l), radius of the pulley (r), and other constants involved in the problem.