# Laws of motion while climbing a Rope

• navneet9431
In summary, the man is able to climb upward due to a force balance of tension acting upward, gravity acting downward, and a ficitious force acting downward due to the acceleration of the frame of reference. When looking through the man's frame, there is no upward reaction force as it is already accounted for in the tension force. The correct kinematic statement is that the sum of forces, including the tension and gravity, equals the mass times the acceleration. This explains how the man is able to climb upward in a state of rest in an accelerated frame.

#### navneet9431

Gold Member
I'm a little confused with the application of laws of motion on a man climbing a rope. Suppose a man of mass Mg is climbing a rope with an acceleration a. Rope is massless. Now if look through the frame of the piece of rope held by the man, there is a force Mg downward by man, ma downward applied by man and T upward. This balances out as the piece is at rest. This equation is correct, though my way of looking at it maybe incorrect. Now if we look through the man's frame, we have Mg downward, T upward, ma upward the reaction by rope, and as we are in an accelerated frame, ma downward. What am I doing wrong and can you explain how is the man able to climb upward i.e. how the forces are acting to give this motion.

In these kinds of problems, you should always draw a force diagram.

In this case, the man is a weight and the rope is assumed to be attached to the ceiling. Gravity acts downward on the man. Rope tension acts upward to counter gravity assuming the rope doesn't break.

Here's a better discussion on it with a force diagram and a hand acting to hold the rope steady. (midway down in the Tension topic)

https://opentextbc.ca/physicstestbook2/chapter/normal-tension-and-other-examples-of-forces/

Is the tension force generated in the Rope,T equal to the force applied by the man?
jedishrfu said:
In these kinds of problems, you should always draw a force diagram.

In this case, the man is a weight and the rope is assumed to be attached to the ceiling. Gravity acts downward on the man. Rope tension acts upward to counter gravity assuming the rope doesn't break.

Here's a better discussion on it with a force diagram and a hand acting to hold the rope steady. (midway down in the Tension topic)

https://opentextbc.ca/physicstestbook2/chapter/normal-tension-and-other-examples-of-forces/

Wait which man or example are you referring to?

In the second example in the link I gave you under tension the man's hand is holding the mass steady and so he is providing an upward force = the force of gravity on the mass.

In the case, a man hanging onto the rope his arms have tension forces to counteract the force of gravity.

jedishrfu said:
Wait which man are you referring to? Man pulls the rope downwards with a force F.So,By Newton's Third Law an upward reaction force F acts on the man.
So is the tension produced in the string equal to F?

#### Attachments

Your forces are off. Think of the man hanging straight down.

There is upward tension in his arms due to gravity pulling down and his arms are hanging onto the rope.

Where his hands hold the rope the is a downward tension force and an upward tension force.

Where the rope is attached to the ceiling there is downward tension in the rope and is an upward force at the ceiling.

Since nothing is moving then the tension forces sum to an upward force that counters the gravitational force pulling the man downward.

If the climber is accelerating relative to Earth at a, the tension in the rope will be M(g + a); his acceleration will add to g. (with the appropriate sign!)

You say that the man experiences three forces: gravity down, tension up, and “ma upward the reaction from the rope”. This is incorrect. The tension is the only upward force. The idea of reaction is that the net force the man applies down on the rope is equal and opposite to the force the rope applies up on the man. This isn’t some separate force. It is a requirement on the forces.

The man has two forces acting on him: gravity downward, and the rope tension upward. The correct kinematic statement is that the SUM of these forces, that is to say, the NET force equals the mass times the acceleration

ΣF = m a

So

Ftension - Fgravity = m a

The force from gravity is m g so

Ftension - m g = m a

And thus

Ftension = m a + mg

navneet9431 said:
Now if we look through the man's frame, we have Mg downward, T upward, ma upward the reaction by rope, and as we are in an accelerated frame, ma downward. What am I doing wrong and can you explain how is the man able to climb upward i.e. how the forces are acting to give this motion.
Attempting to explain the motion of a man using an accelerated frame in which the man is at rest seems like a contradiction in terms. In this frame the man is not moving! The question is how one explains his continued state of being at rest. That's easy. There is a force balance between T upward, mg downward and a ficitious force ma downward where a is not the acceleration of the man but is, instead, the acceleration of the frame of reference.

Of course the man remains at rest. We have chosen a frame where it is so. Of course Newton's second law still applies. We have invented a fictitious force to make it so.

## 1. What are Newton's Laws of Motion?

Newton's Laws of Motion are a set of three physical laws that describe the behavior of objects in motion. They were developed by Sir Isaac Newton in the 17th century and are considered fundamental principles of classical mechanics.

## 2. How do these laws apply to climbing a rope?

The first law, also known as the Law of Inertia, states that an object at rest will remain at rest and an object in motion will remain in motion unless acted upon by an external force. This means that a climber must exert force on the rope in order to move upward, and will continue to move until acted upon by another force, such as friction or gravity.

The second law, also known as the Law of Acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the more force a climber exerts on the rope, the faster they will accelerate upward.

The third law, also known as the Law of Action and Reaction, states that for every action, there is an equal and opposite reaction. This means that the climber's upward force on the rope is met with an equal and opposite force from the rope, allowing them to push off and propel themselves upward.

## 3. How does friction play a role in climbing a rope?

Friction is the force that opposes motion between two surfaces in contact. In the context of climbing a rope, friction is necessary for the climber to grip onto the rope and prevent them from slipping. However, too much friction can also make it difficult for the climber to move upward, so finding the right balance is important.

## 4. Can Newton's laws be applied to different types of rope climbing?

Yes, Newton's laws apply to all forms of rope climbing, whether it be traditional rock climbing, sport climbing, or even rope climbing in a gym. The same principles of forces, acceleration, and friction still apply.

## 5. Are there any other factors that affect climbing a rope besides Newton's laws?

Yes, there are other factors that can affect climbing a rope, such as the weight and strength of the climber, the angle of the rope, and the condition of the rope itself. These factors can impact the amount of force needed to climb and the amount of friction between the climber and the rope.