What is the angle the ladder makes with the ground?

  

A 11.0-kg monkey climbs a uniform ladder with weight w = 1.33 * 102 N and length L = 3.45 m as shown in the figure below. The ladder rests against the wall and makes an angle of = 60.0
with the ground. The upper and lower ends of the ladder rest on
frictionless surfaces. The lower end is connected to the wall by a
horizontal rope that is frayed and can support a maximum tension of only
80.0 N.http://www.webassign.net/serpse8/12-p-024.gifa.) Find the tension in the rope when the monkey is two-thirds of the way up the ladder.

Introduction:
In the field of physics, ladder problems like the one presented in this scenario play a significant role in determining the different forces acting on the ladder and calculating their magnitudes. One such scenario involves a monkey climbing a ladder with a specific weight and length, while the ladder rests against a vertical surface, and one end of the ladder is connected to a wall with a frayed rope. In this scenario, the task involves finding the tension in the rope when the monkey is two-thirds of the way up the ladder.

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Description:
The problem statement details an 11.0-kg monkey climbing a uniform ladder that weighs 1.33 * 10^2 N and measures 3.45 m in length. The ladder leans at an angle of 60.0 degrees with respect to the ground and rests against a vertical wall whose lower end is tied to a rope. The rope has a maximum tension limit of 80.0 N since it is frayed. The objective of the problem is to calculate the tension in the rope when the monkey continues its climb two-thirds of the way up the ladder. Since the upper and lower ends of the ladder rest on frictionless surfaces, the task’s primary purpose is to identify the various forces acting on the ladder, including the gravitational force and tension in the rope, and calculate the required magnitude of the tension force to prevent detachment of the ladder from the wall.

Objectives:

– To apply the principles of equilibrium to solve problems involving ladders and ropes.
– To determine the tension in a rope given the weight and position of a monkey on a ladder.
– To analyze the forces involved in a system consisting of a ladder, a monkey, and a rope connected to a wall.

Learning Outcomes:

By the end of this lesson, students will be able to:

– Explain the concept of equilibrium and how it applies to problems involving ladders and ropes.
– Apply the equations of static equilibrium to calculate the tension in a rope supporting a ladder with a monkey on it.
– Identify the different forces acting on a system consisting of a ladder, a monkey, and a rope connected to a wall, and determine how they affect the system’s equilibrium.
– Solve problems involving ladders, ropes, and monkeys, using a step-by-step approach that involves drawing free-body diagrams, applying equations of static equilibrium, and checking work for consistency and correctness.

Note: The above objectives and learning outcomes are based on the content provided, which is a single problem involving a monkey, a ladder, and a rope. These can be expanded or modified depending on the level and scope of the curriculum.

Solution 1:
To determine the tension in the rope when the monkey is two-thirds of the way up the ladder, we’ll use the principles of equilibrium. When the monkey is at two-thirds of the way up the ladder, the ladder is in static equilibrium with the forces acting on it. Therefore, the sum of the forces in the vertical direction must be zero.

Firstly, we need to find the force exerted by the monkey on the ladder which is equal to its weight.

F_gravity_monkey = m_monkey * g
= 11.0 kg * 9.8 m/s^2
= 107.8 N

Next, we need to calculate the force exerted by the ladder on the wall and the force exerted by the rope on the ladder when the monkey is at two-thirds of the way up the ladder.

F_rope = tension in the rope
= 80.0 N (maximum tension that rope can bear)

We can resolve the weight of the ladder into components parallel and perpendicular to the wall.

F_ladder_parallel = w_ladder * sin(theta)
= (1.33 x 10^2 N) * sin(60.0°)
= 1.1547 x 10^2 N

F_wall = F_ladder_parallel

The sum of the forces in the vertical direction must be zero. Therefore,

F_rope + F_monkey_vertical + F_ladder_vertical = 0
=> F_rope + F_gravity_monkey – F_ladder_perpendicular = 0
=> F_rope + F_gravity_monkey = F_ladder_perpendicular

At two-thirds of the way up the ladder, the length of the ladder that is in contact with the wall is 2L/3 = 2.3 m.

F_ladder_perpendicular = w_ladder * cos(theta) * (2L/3)
= (1.33 x 10^2 N) * cos(60.0°) * (2.3 m / 3.45 m)
= 38.611 N

Therefore,

F_rope + F_gravity_monkey = F_ladder_perpendicular
= 38.611 N

Hence, the tension in the rope when the monkey is two-thirds of the way up the ladder is 69.189 N (i.e., 38.611 N + 107.8 N – 80.0 N).

Solution 2:
Another way to solve this problem is to use the principle of work and energy. When the monkey climbs up the ladder, its potential energy increases, and the work done to increase the potential energy comes from the tension in the rope.

At two-thirds of the way up the ladder, the height of the monkey above the ground is (2/3)Lsin(theta).

The increase in the potential energy of the monkey is given by:

delta_U = m_monkey * g * (2L/3)sin(theta)
= 11.0 kg * 9.8 m/s^2 * (2/3)*3.45*sin(60.0°)
= 191.352 J

The work done by the tension in the rope is given by:

W_rope = F_rope * d = delta_U
=> F_rope = delta_U / d

where d is the distance moved by the monkey.

d = (2/3)Lcos(theta)
= (2/3)*3.45*cos(60.0°)
= 1.72 m

Therefore,

F_rope = delta_U / d
= 191.352 J / 1.72 m
= 111.12 N

However, since the maximum tension that the rope can bear is only 80.0 N, the tension in the rope when the monkey is two-thirds of the way up the ladder is 80.0 N (the maximum tension that the rope can bear).

Suggested Resources/Books:
1. Sears and Zemansky’s University Physics (14th Edition)
2. Physics for Scientists and Engineers with Modern Physics (10th Edition) by Raymond A. Serway and John W. Jewett
3. Fundamentals of Physics Extended (10th Edition) by David Halliday, Robert Resnick and Jearl Walker

Similar Asked Questions:
1. How to calculate tension in a rope in a ladder problem?
2. What is the maximum weight a ladder can support against a wall?
3. How to find the force applied by a ladder when it rests against a wall?
4. What angle should a ladder make with the ground for maximum stability?
5. How to find the horizontal force on a ladder against a wall?A 11.0-kg monkey climbs a uniform ladder with weight w = 1.33 * 102 N and length L = 3.45 m as shown in the figure below. The ladder rests against the wall and makes an angle of = 60.0
with the ground. The upper and lower ends of the ladder rest on
frictionless surfaces. The lower end is connected to the wall by a
horizontal rope that is frayed and can support a maximum tension of only
80.0 N.http://www.webassign.net/serpse8/12-p-024.gifa.) Find the tension in the rope when the monkey is two-thirds of the way up the ladder.

Introduction:
In the field of physics, ladder problems like the one presented in this scenario play a significant role in determining the different forces acting on the ladder and calculating their magnitudes. One such scenario involves a monkey climbing a ladder with a specific weight and length, while the ladder rests against a vertical surface, and one end of the ladder is connected to a wall with a frayed rope. In this scenario, the task involves finding the tension in the rope when the monkey is two-thirds of the way up the ladder.

Description:
The problem statement details an 11.0-kg monkey climbing a uniform ladder that weighs 1.33 * 10^2 N and measures 3.45 m in length. The ladder leans at an angle of 60.0 degrees with respect to the ground and rests against a vertical wall whose lower end is tied to a rope. The rope has a maximum tension limit of 80.0 N since it is frayed. The objective of the problem is to calculate the tension in the rope when the monkey continues its climb two-thirds of the way up the ladder. Since the upper and lower ends of the ladder rest on frictionless surfaces, the task’s primary purpose is to identify the various forces acting on the ladder, including the gravitational force and tension in the rope, and calculate the required magnitude of the tension force to prevent detachment of the ladder from the wall.

Objectives:

– To apply the principles of equilibrium to solve problems involving ladders and ropes.
– To determine the tension in a rope given the weight and position of a monkey on a ladder.
– To analyze the forces involved in a system consisting of a ladder, a monkey, and a rope connected to a wall.

Learning Outcomes:

By the end of this lesson, students will be able to:

– Explain the concept of equilibrium and how it applies to problems involving ladders and ropes.
– Apply the equations of static equilibrium to calculate the tension in a rope supporting a ladder with a monkey on it.
– Identify the different forces acting on a system consisting of a ladder, a monkey, and a rope connected to a wall, and determine how they affect the system’s equilibrium.
– Solve problems involving ladders, ropes, and monkeys, using a step-by-step approach that involves drawing free-body diagrams, applying equations of static equilibrium, and checking work for consistency and correctness.

Note: The above objectives and learning outcomes are based on the content provided, which is a single problem involving a monkey, a ladder, and a rope. These can be expanded or modified depending on the level and scope of the curriculum.

Solution 1:
To determine the tension in the rope when the monkey is two-thirds of the way up the ladder, we’ll use the principles of equilibrium. When the monkey is at two-thirds of the way up the ladder, the ladder is in static equilibrium with the forces acting on it. Therefore, the sum of the forces in the vertical direction must be zero.

Firstly, we need to find the force exerted by the monkey on the ladder which is equal to its weight.

F_gravity_monkey = m_monkey * g
= 11.0 kg * 9.8 m/s^2
= 107.8 N

Next, we need to calculate the force exerted by the ladder on the wall and the force exerted by the rope on the ladder when the monkey is at two-thirds of the way up the ladder.

F_rope = tension in the rope
= 80.0 N (maximum tension that rope can bear)

We can resolve the weight of the ladder into components parallel and perpendicular to the wall.

F_ladder_parallel = w_ladder * sin(theta)
= (1.33 x 10^2 N) * sin(60.0°)
= 1.1547 x 10^2 N

F_wall = F_ladder_parallel

The sum of the forces in the vertical direction must be zero. Therefore,

F_rope + F_monkey_vertical + F_ladder_vertical = 0
=> F_rope + F_gravity_monkey – F_ladder_perpendicular = 0
=> F_rope + F_gravity_monkey = F_ladder_perpendicular

At two-thirds of the way up the ladder, the length of the ladder that is in contact with the wall is 2L/3 = 2.3 m.

F_ladder_perpendicular = w_ladder * cos(theta) * (2L/3)
= (1.33 x 10^2 N) * cos(60.0°) * (2.3 m / 3.45 m)
= 38.611 N

Therefore,

F_rope + F_gravity_monkey = F_ladder_perpendicular
= 38.611 N

Hence, the tension in the rope when the monkey is two-thirds of the way up the ladder is 69.189 N (i.e., 38.611 N + 107.8 N – 80.0 N).

Solution 2:
Another way to solve this problem is to use the principle of work and energy. When the monkey climbs up the ladder, its potential energy increases, and the work done to increase the potential energy comes from the tension in the rope.

At two-thirds of the way up the ladder, the height of the monkey above the ground is (2/3)Lsin(theta).

The increase in the potential energy of the monkey is given by:

delta_U = m_monkey * g * (2L/3)sin(theta)
= 11.0 kg * 9.8 m/s^2 * (2/3)*3.45*sin(60.0°)
= 191.352 J

The work done by the tension in the rope is given by:

W_rope = F_rope * d = delta_U
=> F_rope = delta_U / d

where d is the distance moved by the monkey.

d = (2/3)Lcos(theta)
= (2/3)*3.45*cos(60.0°)
= 1.72 m

Therefore,

F_rope = delta_U / d
= 191.352 J / 1.72 m
= 111.12 N

However, since the maximum tension that the rope can bear is only 80.0 N, the tension in the rope when the monkey is two-thirds of the way up the ladder is 80.0 N (the maximum tension that the rope can bear).

Suggested Resources/Books:
1. Sears and Zemansky’s University Physics (14th Edition)
2. Physics for Scientists and Engineers with Modern Physics (10th Edition) by Raymond A. Serway and John W. Jewett
3. Fundamentals of Physics Extended (10th Edition) by David Halliday, Robert Resnick and Jearl Walker

Similar Asked Questions:
1. How to calculate tension in a rope in a ladder problem?
2. What is the maximum weight a ladder can support against a wall?
3. How to find the force applied by a ladder when it rests against a wall?
4. What angle should a ladder make with the ground for maximum stability?
5. How to find the horizontal force on a ladder against a wall?

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