If at any time t the length of the rope hanging from the table is X then the force acting on the rope is gravitational force acting on the portion of rope hanging from the table as given by
This is equal to the mass of the rope times acceleration of the rope at that time which is given by
So the equation of motion is given by
This is the equation of motion.
b) if a larger part of the rope hangs from the table so the amount gravitational force on that part will increase that will also increase the acceleration of the rope.
c) let is a solution of our problem. So we can use this solution to the equation of motion as shown below
So we see that we have two values of c one is positive and other is negative for any value of b.
d) the general solution is the sum of two different solutions for two values of c as given below
Where and are constant.
This is also a solution of equation of motion as it satisfies equation of motion as shown below
So we see that sum of any two solutions is also a solution of equation of motion.
e) so for this we find the velocity of the rope is given by
From the question at t = 0 velocity was zero because the rope was at rest.
Using this condition we get
is a arbitrary constant.
So the solution is given below
.
18 A flexible rope of mass m and length L slides without friction over the edge...
18 A flexible rope of mass m and length L slides without friction over the edge of a table. Let x be the length of the rope that is hanging over the edge at a given moment in time (a) Show that r satisfies the equation of motion/dt2 -gr/L. Hint: Use F-dp/ dt, which allows you to handle the two parts of the rope separately even though mass is moving out of one part and into the other (b) Give...
A heavy rope with length L and mass M is attached to the ceiling and is allowed to hang freely. (a) Find an expression for the tension in the rope at a point a distance y from the bottom, and use this to show that the speed of transverse waves on the rope is independent of its mass and length but does depend on the distance y according to the equation ?=??. (b) If L = 3.0 m and the...
L-M 7. A uniform flexible chain of length L, with mass per unit length A, passes over a small, frictionless, massless pulley. It is released from a rest position with a length of chain y hanging from one side and a length l-y from the other side. Find the acceleration a as a function of y a(y)-?
L-M 7. A uniform flexible chain of length L, with mass per unit length A, passes over a small, frictionless, massless pulley. It is released from a rest position with a length of chain y hanging from one side and a length l-y from the other side. Find the acceleration a as a function of y a(y)-?
A uniform flexible chain of length L, with mass per unit length A, passes over a small (radius is negligible), frictionless pulley, as shown in figure 1. One side of the chain with length x is tied to a block with mass m. The chain and block are released from a rest position at t = 0 s. (Hint: the chain can be treated as a rope with non-negligible mass)a. Find x such that the chain and the block are...
A rod of length L, mass m , and R resistance slides down with no friction. A set of two conducting, parallel railings with negligible resistance (held at an angle to the ground). A constant magnetic field B is present everywhere and is directed upwards (perpendicular to the ground) For the following values: What is the velocity of the rod at equilibrium ? (g=10m/s^2) We were unable to transcribe this imageWe were unable to transcribe this image
A rod of length L, mass m , and R resistance slides down with no friction. A set of two conducting, parallel railings with negligible resistance (held at an angle to the ground). A constant magnetic field B is present everywhere and is directed upwards (perpendicular to the ground) For the following values: What is the velocity of the rod at equilibrium ? (g=10m/s^2) We were unable to transcribe this imageWe were unable to transcribe this image
A compound pendulum is made up of a rod of length L, with mass M and a solid sphere of radius r, with mass m (see figure below). The pendulum is pivoted about one end and released from rest from and angle of 0, (angle with the vertical). (a) Find the distance, dom, of center of mass of this pendulum from its pivot. (b) Draw a free body diagram and write down Newton's 2nd Law (for rotation) for the pendulum...
An ensemble of edge dislocations present in a bulk specimen of mass 'M has a total length of 'L.' The physical density of the material is 'r'. All the dislocations are present in parallel slip planes and they have Burgers vector of b. However, it is noted that only 30% of the dislocations are sessile in nature and the rest are mobile. a) What would be the shear strain rate if the mobile (5) b) To create a shear strain...
Example 24 A beam of length L and mass m, is pivoted at one end and suspended from a spring of stiffness k distance L from the pivot. A dashpot is also positioned a distance R from the pivot. Show how the equation of motion below can be derived. de 3cR2 de dt 3kLi 0 = 0 mL2 mL dt where x is the displacement at the free end, and c is the damping coefficient (Ns/m) Example 24 A beam...