According to the potential-energy criterion for equilibrium, what mathematical condition must be met for a frictionless connected system with one degree of freedom to be in equilibrium?
Explanation
The potential-energy method provides a powerful way to find equilibrium positions. An equilibrium state corresponds to a stationary point (peak, valley, or inflection point) on the potential energy graph, where the slope (first derivative) is zero.
Other questions
According to the principle of virtual work, what is the condition for a rigid body subjected to a coplanar force system to be in equilibrium?
What is a 'virtual displacement' in the context of the principle of virtual work?
When applying the virtual-work equation to a rigid body, why is the work done by internal forces not included?
What does the concept of 'degrees of freedom' signify for a system of connected rigid bodies?
When is the work done by a force or couple moment considered positive in the context of virtual work?
What are the two examples of conservative forces discussed in Chapter 11.4?
How is the potential energy function V for a system defined?
What is the condition for an equilibrium configuration to be defined as 'unstable' using the potential energy method?
In Example 11.1, a two-member linkage is analyzed for equilibrium. Each member has a mass of 10 kg, and a horizontal force of 25 N is applied. What is the equilibrium angle theta?
For the simply supported beam in Fig. 11-4, a downward vertical force P is applied at the center, and the beam has a total length l. If a virtual rotation d(theta) is given about point B, what is the virtual work equation?
In the scissors linkage of Example 11.2, a force P is required to maintain equilibrium at theta = 60 degrees. The spring has stiffness k = 5000 N/m and is unstretched at theta = 30 degrees. What is the required force P?
In Example 11.3, a mechanism supports a 10 kg box. What is the couple moment M needed to maintain equilibrium when theta = 60 degrees?
For the mechanism in Example 11.4 supporting a 50-lb cylinder, the angle theta for equilibrium is found by solving the equation '800 sin(theta)cos(theta) - 800 sin(theta) + 100 cos(theta) = 0'. What is the value of theta?
What type of force is friction, and why is its work not typically included when using the potential-energy method?
If a body of weight W is located a distance y above a chosen datum, what is its gravitational potential energy Vg?
A spring with stiffness k is stretched by an amount s from its unstretched position. What is its elastic potential energy Ve?
In Example 11.5, a uniform link with mass 10 kg and length 0.6 m is connected to a spring with k = 200 N/m. The system has two equilibrium positions, one at theta = 0 degrees and another at theta = 53.8 degrees. What is the stability of the equilibrium at theta = 53.8 degrees?
What defines a 'neutral equilibrium' configuration for a system?
For the system in Example 11.6, a 1.5 Mg load is on a platform supported by a spring with stiffness 18 kN/m. The spring is unstretched at theta = 60 degrees. What is the stability of the equilibrium position at theta = 28.18 degrees?
A uniform block of mass m and height h rests on top of a half cylinder of radius R, as shown in Example 11.7. What is the condition for this equilibrium to be unstable?
In the 'Procedure for Analysis' using the virtual-work equation, what is the first step after drawing the free-body diagram of the system?
In the 'Procedure for Analysis' using potential-energy methods, what is the step to find the equilibrium position of the system?
For the linkage in problem F11-1, each link is 1.5 m long and has a mass of 20 kg. What is the required magnitude of the vertical force P to maintain equilibrium at theta = 60 degrees?
A 50-kg smooth rod is held in equilibrium at theta = 60 degrees by a horizontal force P, as shown in problem F11-2. The rod has a length of 5 m. What is the magnitude of force P?
What does the work of a conservative force depend on?
For the linkage in problem F11-4, a force P = 6 kN is applied. The spring has k = 20 kN/m and is unstretched at theta = 60 degrees. What is the angle theta for equilibrium?
A 50-kg bar is in equilibrium as shown in problem F11-5. The spring is unstretched at theta = 60 degrees and has a stiffness of k = 600 N/m. What is the angle theta for equilibrium?
For the scissors linkage in problem F11-6 subjected to a force P = 150 N, the spring has k = 15 kN/m and is unstretched at theta = 0 degrees. What is the angle theta for equilibrium?
If the potential energy V of a one-degree-of-freedom system is given by V = (4x^3 - x^2 - 3x + 10) ft-lb, what is an equilibrium position for the system?
What is the relationship between the work U_1-2 done by conservative forces on a system and its potential function V as it moves from state 1 to state 2?
Which statement correctly describes the stability condition based on the second derivative of the potential energy function, d²V/dq²?
In problem 11-5, a 10 lb uniform rod AB is in equilibrium. What is the required force developed in the spring to keep the rod in equilibrium when theta = 35 degrees?
What is the primary advantage of using the principle of virtual work for solving equilibrium problems involving systems of connected bodies?
If a force or couple moment acts on a body but its point or line of action does not move during a virtual displacement, what is the virtual work done by that force or moment?
For the truck on the inspection scale in problem 11-14, a known mass m is placed a distance s from fulcrum B. The truck has an unknown mass m_t with its center of gravity a distance d from C. When empty, the scale is balanced. Determine the mass of the truck, m_t.
What is the condition for neutral equilibrium if the first and second derivatives of the potential energy function are zero?
In problem 11-19, a spring is unstretched when theta = 45 degrees and has a stiffness of k = 1000 lb/ft. Each of the two cylinders weighs 50 lb. Determine the angle theta for equilibrium.
For the toggle press in problem 11-9, a force P = 100 N is applied to the lever arm. What is the clamping force F developed in the block when theta = 45 degrees?
A block of weight W is supported by a spring. The potential function is given by V = -Wy + (1/2)ky^2, where y is measured downwards from the spring's unstretched position. What is the equilibrium position, y_eq?
The vent plate in problem 11-21 weighs 15 lb with its center of gravity at G. The spring is unstretched when theta = 0 degrees. What is the required stiffness k of the spring for the plate to be in equilibrium at theta = 30 degrees?
A 20-kg cylinder is supported by a mechanism with a smooth peg at B that can slide freely in a slot, as shown in problem 11-24. What is the magnitude of the couple moment M required to maintain equilibrium at theta = 30 degrees?
What does a zero second derivative of the potential energy function (d²V/dq² = 0) at an equilibrium point signify?
For the crankshaft in problem 11-25, a torque M = 50 lb-ft is applied. The connecting rod is 5 in. and the crank radius is 3 in. What is the vertical compressive force F on the piston for equilibrium when theta = 60 degrees?
If a potential energy function for a system is V = (3y^3 + 2y^2 - 4y + 50) J, what is the stability of the equilibrium position at y = 2/3 m?
What is the primary difference between the 'method of virtual displacements' and the 'method of virtual forces'?
For the lift table in problem 11-1, a 200-kg crate is at the position theta = 30 degrees. Neglecting the mass of the lift table components, what is the force in the hydraulic cylinder AD?
What is the key assumption made about the bodies in Chapter 11 that allows the work of internal forces to be ignored?
In the 'Procedure for Analysis' using virtual work, how should the position coordinates 's' be defined?
The potential energy of a one-degree-of-freedom system is V = (24 sin(theta) + 10 cos(2*theta)) ft-lb. Which of the following is an equilibrium position?