A rope which is 7 meters long is laid out on a horizontal table.
(a) Assuming that the coefficient of static friction between the rope and the top of the table is 0.5, show that the rope will begin to slide if at least 3 meters of it hangs off the edge.
(b) Once it starts to slide, the coefficient of kinetic friction is 0.1. Find the speed of the rope as the last bit of it slides off the table.
To determine if the rope will begin to slide, we need to compare the force of friction acting on the rope to the maximum static frictional force that can be exerted.
(a) The force of friction can be calculated using the formula:
Frictional force = coefficient of static friction * normal force
In this case, the normal force is equal to the weight of the hanging part of the rope, which is given by:
Weight = mass * gravitational acceleration
Since the rope is laid out horizontally, only a portion of it is hanging off the edge, which we need to consider. Let's call the length of the hanging part of the rope "x".
Weight = mass * gravitational acceleration = (mass per unit length * length) * gravitational acceleration = (mass per unit length * x) * gravitational acceleration
The maximum static frictional force that can be exerted can be calculated using:
Maximum static frictional force = coefficient of static friction * normal force
Maximum static frictional force = coefficient of static friction * [(mass per unit length * x) * gravitational acceleration]
The rope will just begin to slide when the force of friction is equal to the maximum static frictional force. Therefore:
Frictional force = Maximum static frictional force
Coefficient of static friction * [(mass per unit length * x) * gravitational acceleration] = coefficient of static friction * normal force
Simplifying the equation:
Coefficient of static friction * (mass per unit length * x) * gravitational acceleration = coefficient of static friction * [(mass per unit length * x) * gravitational acceleration]
The coefficients of static friction and gravitational acceleration cancel out, resulting in:
mass per unit length * x = mass per unit length * x
Since the mass per unit length is the same throughout the rope, we can cancel it out as well:
x = x
This equation shows that the length of the hanging rope, "x", is unchanged. In other words, as long as the hanging length of the rope is at least 3 meters, it will begin to slide.
(b) Once the rope starts to slide, we can calculate its speed as the last bit of it slides off the table using the principles of conservation of energy. Since there is no friction when the rope is in motion, the gravitational potential energy of the hanging part of the rope is converted into kinetic energy.
The potential energy of the hanging part of the rope is given by:
Potential energy = mass * gravitational acceleration * height
Since the length of the rope that is hanging off the table is decreasing as it slides, the height is changing. Let's call the length of the hanging part of the rope when it has fully slid off the table "y". The height in this case will be y.
Potential energy = (mass per unit length * y) * gravitational acceleration * y
This potential energy is converted entirely into kinetic energy as the rope slides off the table. The kinetic energy is given by:
Kinetic energy = (1/2) * mass * velocity^2
We need to find the velocity of the rope when it fully slides off, so we equate potential energy and kinetic energy:
(mass per unit length * y) * gravitational acceleration * y = (1/2) * mass * velocity^2
At this point, we can cancel out the mass per unit length and the mass terms, as they are the same throughout the rope:
y * gravitational acceleration * y = (1/2) * velocity^2
Simplifying the equation:
y^2 * gravitational acceleration = (1/2) * velocity^2
Rearranging the equation to solve for velocity:
velocity^2 = 2 * y^2 * gravitational acceleration
Taking the square root of both sides to find velocity:
velocity = sqrt(2 * y^2 * gravitational acceleration)
Substituting y = 4 meters (since the hanging length of the rope in this case is 4 meters):
velocity = sqrt(2 * (4^2) * gravitational acceleration)
velocity = sqrt(2 * 16 * gravitational acceleration)
velocity = sqrt(32 * gravitational acceleration)
Finally, we can substitute the value of gravitational acceleration (approximately 9.8 m/s^2):
velocity = sqrt(32 * 9.8)
velocity ≈ 17.49 m/s
Therefore, the speed of the rope as the last bit of it slides off the table will be approximately 17.49 meters per second.