Skip to main content
Ch 07: Newton's Third Law
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 07: Newton's Third LawProblem 21
Chapter 7, Problem 21

A mobile at the art museum has a 2.0 kg steel cat and a 4.0 kg steel dog suspended from a lightweight cable, as shown in FIGURE EX7.21. It is found that θ1\(\theta\)_1 = 20° when the center rope is adjusted to be perfectly horizontal. What are the tension and the angle of rope 3?

Verified step by step guidance
1
Step 1: Analyze the forces acting on the system. The mobile is in equilibrium, meaning the sum of forces in both the horizontal and vertical directions must be zero. Identify the tensions in the ropes (T₁, T₂, T₃, T₄, T₅) and the angles θ₁ and 25° as given in the diagram.
Step 2: Write the equations for vertical force equilibrium. The vertical components of T₃ and T₁ must balance the weights of the steel cat (2.0 kg) and steel dog (4.0 kg). Use the equation: T₃sin(25°) + T₁sin(θ₁) = (2.0 kg + 4.0 kg) * g, where g is the acceleration due to gravity (9.8 m/s²).
Step 3: Write the equations for horizontal force equilibrium. The horizontal components of T₃ and T₁ must balance each other since the center rope is perfectly horizontal. Use the equation: T₃cos(25°) = T₁cos(θ₁).
Step 4: Solve the system of equations. Use the two equations derived in steps 2 and 3 to solve for T₁ and θ₁. This involves substituting one equation into the other and isolating the variables.
Step 5: Interpret the results. Once T₁ and θ₁ are determined, verify that they satisfy both the vertical and horizontal equilibrium conditions. This ensures the solution is consistent with the physical setup of the problem.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
21m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tension in a Rope

Tension is the force exerted along a rope or cable when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the ropes must balance the gravitational forces acting on the suspended weights (the cat and the dog). Understanding how to calculate tension is crucial for solving problems involving static equilibrium.
Recommended video:
Guided course
06:34
Calculating Tension in a Pendulum with Energy Conservation

Static Equilibrium

Static equilibrium occurs when an object is at rest and the sum of all forces and torques acting on it is zero. In the context of the mobile, both the vertical and horizontal components of the forces must balance out. This principle allows us to set up equations based on the weights of the cat and dog and the angles of the ropes to find unknown tensions and angles.
Recommended video:
Guided course
08:11
Static Friction & Equilibrium

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In this problem, these functions are essential for resolving the tension forces into their vertical and horizontal components based on the angles θ1 and θ3. This resolution is necessary to apply the conditions of static equilibrium effectively.
Recommended video:
Guided course
08:30
Intro to Wave Functions
Related Practice
Textbook Question

A 2.0-m-long, 500 g rope pulls a 10 kg block of ice across a horizontal, frictionless surface. The block accelerates at 2.0 m/s2. How much force pulls forward on he rope? Assume that the rope is perfectly horizontal.

1
views
Textbook Question

The 100 kg block in FIGURE EX7.24 takes 6.0 s to reach the floor after being released from rest. What is the mass of the block on the left? The pulley is massless and frictionless.

2
views
Textbook Question

A 500 kg air conditioner sits on the flat roof of a building. The coefficient of static friction between the roof and the air conditioner is 0.90. A massless rope attached to the air conditioner passes over a massless, frictionless pulley at the edge of the roof. In an effort to drag the air conditioner to the edge of the roof, four 100 kg students hang from the free end of the rope, but the air conditioner refuses to budge. What is the magnitude of the rope tension at the point where it is attached to the air conditioner?

2
views
Textbook Question

Two blocks are attached to opposite ends of a massless rope that goes over a massless, frictionless, stationary pulley. One of the blocks, with a mass of 6.0 kg, accelerates downward at (3/4)g. What is the mass of the other block?

2
views
Textbook Question

FIGURE EX7.17 shows two 1.0 kg blocks connected by a rope. A second rope hangs beneath the lower block. Both ropes have a mass of 250 g. The entire assembly is accelerated upward at 3.0 m/s2 by force F. What is the tension at the top end of rope 1?

14
views
Textbook Question

The 1.0 kg block in FIGURE EX7.23 is tied to the wall with a rope. It sits on top of the 2.0 kg block. The lower block is pulled to the right with a tension force of 20 N. The coefficient of kinetic friction at both the lower and upper surfaces of the 2.0 kg block is μk = 0.40. What is the tension in the rope attached to the wall?

2
views