Tension in ropes used to build the pyramids

• Priscilla
In summary, the tension in the rope is required to keep the block moving up the ramp at a constant speed, and it is dependent on the angle of the ramp.
Priscilla

Homework Statement

A mass M is hauled from ground level up an inclined plane that makes an angle 0 with the horizontal by means of a rope passing over a frictionless pulley. The mass is pulled along until it reaches a height of H. The building of the Egyptian pyramids and many other ancient construction jobs used such ramps. Logs were sometimes used as rollers to reduce friction.

a)Show that if the contact between the mass and ramp is frictionless, the work done by the tension in the rope (or equivalently, by the person hauling on the rope) is independent of the angle 0 .
b)Calculate the work done by the tension in the rope as a function of the ramp angle if the coefficient of kinetic friction between the mass and the surface is U_k.

Homework Equations

W=Fd
f_k = U_k*N
( N = normal force f_k = kinetic friction)

The Attempt at a Solution

6a, I have no idea how to prove it. Because I thought the tension is dependent of the angle. The angle is used to calculate the F_N, Fs...

6b
W=Fd d = H
N=mg cos 0
F=U_k mg cos 0
W=U_k mg cos 0 H

For a), what is the tension required just to keep the block moving (no acceleration)? (hint: think about what would it be if the ramp was perfectly horizontal; perfectly vertical - the answer is somewhere in between).

For b) do a free body diagram with the block moving up the ramp at constant speed. What must the tension in the rope be equal to if there is no acceleration? (hint: in order to work out the tension you have to determine the normal force, which is a function of the ramp angle.).

AM

Priscilla said:
(Private message).

For 6a) The way you tell me sound like it's dependent of the angle, but the question is asking to prove that the work done by tension is independent of the angle.

For 6b) The equations that I used are not correct?
You should use the thread so others can benefit.

Use the definition of work: $W = \vec{F}\cdot\vec{S}$. What is F (tension)? (hint: it is a function of $\theta$). What is the distance (hint: express in terms of h and $\theta$)? Multiply them together and see if the result involves $\theta$

b).Your equations are partially correct. You are missing the work done against the force of gravity.

You have to express the tension in terms of the other forces that balance the tension. Those forces are gravity and friction. Use a free body diagram and work them out.

AM

For a)
Tension on the inclined plane = Mg sin [0]
Distance of the inclined plane = H/ sin[0]
W = Fd = Td = (Mg sin[0]) (H/ sin[0]) = MgH

For b)
f_k = u_k N
N = Mg cos [0]
F = u_k Mg cos [0]
W = Fd
d = H/sin[0]
W = (u_k Mg cos [0] H)/ (sin[0])

Are they correct?

Priscilla said:
For a)
Tension on the inclined plane = Mg sin [0]
Distance of the inclined plane = H/ sin[0]
W = Fd = Td = (Mg sin[0]) (H/ sin[0]) = MgH

For b)
f_k = u_k N
N = Mg cos [0]
F = u_k Mg cos [0]
W = Fd
d = H/sin[0]
W = (u_k Mg cos [0] H)/ (sin[0])

Are they correct?
a) is correct. But not b). Compare your answer in b) to that in part a). Why would the work be LESS if you add friction?

AM

W = MgH + (u_k Mg cos [0] H)/ (sin[0])?

Priscilla said:
W = MgH + (u_k Mg cos [0] H)/ (sin[0])?
Looks right.

$$Tension = mg\sin\theta + \mu_k mg\cos\theta$$

$$W = F\cdot s =\frac{(mg\sin\theta + \mu_k mg\cos\theta)h}{\sin\theta}= mgh\left(1 + \frac{\mu_k}{\tan\theta\right)}$$

AM

Thanks!

1. How were the ropes used to build the pyramids able to withstand the tension?

The ropes used to build the pyramids were made of natural fibers such as papyrus, flax, or palm leaves. These fibers were twisted together to create a stronger and more durable rope. Additionally, the ropes were coated in a resin or animal fat to increase their strength and flexibility. This combination of materials and techniques allowed the ropes to withstand the tension of the heavy stones used in the construction of the pyramids.

2. How were the ropes attached to the stones in the pyramid?

The ropes were tied to the stones using a variety of knots such as the clove hitch or the timber hitch. These knots were specifically chosen for their ability to hold tension and distribute weight evenly. The ropes were also sometimes wrapped around the stones multiple times to increase their strength and stability.

3. How did the ancient Egyptians transport the heavy stones using ropes?

The ancient Egyptians used a technique called "levering" to transport the heavy stones using ropes. This involved placing a large wooden beam under the stone and using shorter beams or ropes to create a lever system. This allowed the workers to lift and move the stones with relative ease, using the ropes to control the direction and placement of the stones.

4. Did the ancient Egyptians have any special tools to help them with the ropes?

Yes, the ancient Egyptians had a variety of tools to help them with the ropes used in building the pyramids. These included wooden pulleys, ramps, rollers, and sledges. These tools were used to lift, move, and position the heavy stones, making it easier for the workers to manipulate the ropes and control the tension.

5. How long did the ropes used to build the pyramids last?

The lifespan of the ropes used to build the pyramids varied depending on the materials used and the conditions they were exposed to. Some ropes may have only lasted a few years, while others could have lasted for decades. However, due to the harsh climate of Egypt, it is likely that the ropes needed to be replaced frequently, and the ancient Egyptians would have had to constantly maintain and repair them in order to keep the tension in the ropes strong enough for construction.

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