Explanation Given information: The system is in equilibrium when θ 1 = θ 2 = 0 . Calculation: Show the free-body diagram of the arrangement as in Figure 1. Consider the movement of spring at left end is from a to b , and the right end is from a ′ to b ′ . Find the elongation of the spring ( s ) using the relation. s = a ′ b ′ − a b = r θ 1 − r θ 2 = r ( θ 1 − θ 2 ) Find the potential energy ( V ) using the relation. V = V g + V s = 1 2 k s 2 + P l cos θ 1 − W l cos θ 2 = 1 2 k r 2 ( θ 1 − θ 2 ) 2 + P l cos θ 1 − W l cos θ 2 (1) Here, the spring constant is k . Differentiate the Equation (1) with respect to θ 1 . ∂ V ∂ θ 1 = 2 × 1 2 k r 2 ( θ 1 − θ 2 ) − P l sin θ 1 − 0 = k r 2 ( θ 1 − θ 2 ) − P l sin θ 1 (2) Differentiate the Equation (2) with respect to θ 1 . ∂ 2 V ∂ θ 1 2 = k r 2 − P l cos θ 1 Differentiate the equation (2) with θ 2 to find the derivative of ∂ 2 V ∂ θ 1 ∂ θ 2 . ∂ 2 V ∂ θ 1 ∂ θ 2 = − k r 2 Differentiate the Equation (1) with respect to θ 2 . ∂ V ∂ θ 2 = − 2 × 1 2 k r 2 ( θ 1 − θ 2 ) + 0 − W l sin θ 2 = − k r 2 ( θ 1 − θ 2 ) − W l sin θ 2 (3) Differentiate the Equation (3) with respect to ∂ θ 2 . ∂ 2 V ∂ θ 2 2 = k r 2 + W l cos θ 2 Condition 1: When the equilibrium is stable, θ 1 = θ 2 = 0 . Substitute 0 for θ 1 and 0 for θ 2 in Equation (2). ∂ V ∂ θ 1 = k r 2 ( 0 − 0 ) − P l sin 0 = 0 Substitute 0 for θ 1 and 0 for θ 2 in Equation (3). ∂ V ∂ θ 2 = − k r 2 ( 0 − 0 ) − W l sin 0 = 0 ∂ V ∂ θ 1 = ∂ V ∂ θ 2 = 0 The condition is satisfied

Vector Mechanics for Engineers: Statics, 11th Edition

11th Edition

ISBN: 9780077687304

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek

Publisher: McGraw-Hill Education

See similar textbooks

Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

Videos

Question

Chapter 10.2, Problem 10.99P

To determine

Find the range of values of P for which the equilibrium is stable.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.2, Problem 10.98P

Chapter 10.2, Problem 10.100P

Students have asked these similar questions

How do i solve this problem?

Q4/ A compressor is driven motor by mean of a flat belt of thickness 10 mm and a width of 250 mm. The motor pulley is 300 mm diameter and run at 900 rpm and the compressor pulley is 1500 mm diameter. The shaft center distance is 1.5 m. The angle of contact of the smaller pulley is 220° and on the larger pulley is 270°. The coefficient of friction between the belt and the small pulley is 0.3, and between the belt and the large pulley is 0.25. The maximum allowable belt stress is 2 MPa and the belt density is 970 kg/m³. (a) What is the power capacity of the drive and (b) If the small pulley replaced by V-grooved pulley of diameter 300 mm, grooved angle of 34° and the coefficient of friction between belt and grooved pulley is 0.35. What will be the power capacity in this case, assuming that the diameter of the large pulley remain the same of 1500 mm.

You are tasked with designing a power drive system to transmit power between a motor and a conveyor belt in a manufacturing facility as illustrated in figure. The design must ensure efficient power transmission, reliability, and safety. Given the following specifications and constraints, design drive system for this application: Specifications: Motor Power: The electric motor provides 10 kW of power at 1,500 RPM. Output Speed: The output shaft should rotate at 150 rpm. Design Decisions: Transmission ratio: Determine the necessary drive ratio for the system. Shaft Diameter: Design the shafts for both the motor and the conveyor end. Material Selection: Choose appropriate materials for the gears, shafts. Bearings: Select suitable rolling element bearings. Constraints: Space Limitation: The available space for the gear drive system is limited to a 1-meter-long section. Attribute 4 of CEP Depth of knowledge required Fundamentals-based, first principles analytical approach…

Chapter 10 Solutions

Vector Mechanics for Engineers: Statics, 11th Edition

Ch. 10.1 - 10.1 Determine the vertical force P that must be...Ch. 10.1 - Determine the horizontal force P that must be...Ch. 10.1 - Prob. 10.3P Ch. 10.1 - 10.3 and 10.4 Determine the couple M that must be...Ch. 10.1 - A spring of constant 15 kN/m connects points C and...Ch. 10.1 - A spring of constant 15 kN/m connects points C and...Ch. 10.1 - The two-bar linkage shown is supported by a pin...Ch. 10.1 - Prob. 10.8P Ch. 10.1 - Prob. 10.9P Ch. 10.1 - Prob. 10.10P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.