Solved: If the potential function for a conservative

The scissors jack supports a load P. Determine the axial force in the screw necessary for equilibrium when the jack is in the position \(\theta\). Each of the four links has a length L and is pin connected at its center. Points B and D can move horizontally.

Use the method of virtual work to determine the tension in cable AC. The lamp weighs 10 lb.

The vent plate is supported at B by a pin. If it weighs 15 lb and has a center of gravity at G, determine the stiffness k of the spring so that the plate remains in equilibrium at \(\theta = 30^{\circ}\). The spring is unstretched when \(\theta = 0^{\circ}\).

The spring has an unstretched length of 0.3 m. Determine the angle \(\theta\) for equilibrium if the uniform links each have a mass of 5 kg.

The pin-connected mechanism is constrained at A by a pin and at B by a roller. If P = 10 lb, determine the angle \(\theta\) for equilibrium. The spring is unstretched when \(\theta = 45^{\circ}\). Neglect the weight of the members.

The pin-connected mechanism is constrained by a pin at A and a roller at B. Determine the force P that must be applied to the roller to hold the mechanism in equilibrium when \(\theta = 30^{\circ}\). The spring is unstretched when \(\theta = 45^{\circ}\). Neglect the weight of the members.

The 4-ft members of the mechanism are pin connected at their centers. If vertical forces \(P_1 = P_2 = 30\ lb\) act at C and E as shown, determine the angle \(\theta\) for equilibrium. The spring is unstretched when \(\theta = 45^{\circ}\). Neglect the weight of the members.

If each of the three links of the mechanism has a weight of 20 lb, determine the angle \(\theta\) for equilibrium of the spring, which, due to the roller guide, always remains horizontal and is unstretched when \(\theta = 0^{\circ}\).

When \(\theta = 20^{\circ}\), the 50-lb uniform block compresses the two vertical springs 4 in. If the uniform links AB and CD each weigh 10 lb, determine the magnitude of the applied couple moments M needed to maintain equilibrium when \(\theta = 20^{\circ}\).

The thin rod of weight W rest against the smooth wall and floor. Determine the magnitude of force P needed to hold it in equilibrium for a given angle \(\theta\).

Determine the force F acting on the cord which is required to maintain equilibrium of the horizontal 10-kg bar AB. Hint: Express the total constant vertical length l of the cord in terms of position coordinates \(s_1\) and \(s_2\). The derivative of this equation yields a relationship between \(\delta_1\) and \(\delta_2\).

Determine the angles \(\theta\) for equilibrium of the 4-lb disk using the principle of the virtual work. Neglect the weight of the rod. The spring is unstretched when \(\theta = 0^{\circ}\) and always remains in the vertical position due to the roller guide.

Each member of the pin-connected mechanism has a mass of 8 kg. If the spring is unstretched when \(\theta = 0^{\circ}\), determine the angle \(\theta\) for equilibrium. Set \(k = 2500\ N/m\) and \(M = 50\ N \cdot m\).

Each member of the pin-connected mechanism has a mass of 8 kg. If the spring is unstretched when \(\theta = 0^{\circ}\), determine the required stiffness k so that the mechanism is in equilibrium when \(\theta = 30^{\circ}\). Set M = 0.

The service window at a fast-food restaurant consists of glass doors that open and close automatically using a motor which supplies a torque M to each door. The far end, A and B, move along the horizontal guides. If a food tray becomes stuck between the doors as shown, determine the horizontal force the doors exert on the tray at the position \(\theta\).

If the spring is unstretched when \(\theta = 30^{\circ}\), the mass of the cylinder is 25 kg, and the mechanism is in equilibrium when \(\theta = 45^{\circ}\), determine the stiffness k of the spring. Rod AB slides freely through the collar at A. Neglect the mass of the rods.

If the spring has a torsional stiffness of \(k = 300\ N \cdot m / rad\) and it is unstretched when \(\theta = 90^{\circ}\), determine the angle \(\theta\) when the frame is in equilibrium.

A 5-kg uniform serving table is supported on each side by pairs of two identical links, AB and CD, and springs CE. If the bowl has a mass of 1 kg, determine the angle \(\theta\) where the table is in equilibrium. The springs each have a stiffness of \(k = 200\ N/m\) and are unstretched when \(\theta = 90^{\circ}\). Neglect the mass of the links.

A 5-kg uniform serving table is supported on each side by two pairs of identical links, AB and CD, and springs CE. If the bowl has a mass of 1 kg and is in equilibrium when \(\theta = 45^{\circ}\), determine the stiffness k of each spring. The springs are unstretched when \(\theta = 90^{\circ}\). Neglect the mass of the links.

Determine the weight of block G required to balance the differential lever when the 20-lb load F is placed on the pan. The lever is in balance when the load and block are not on the lever. Take x = 12 in.

If the load F weighs 20 lb and the block G weighs 2 lb, determine its position x for equilibrium of the differential lever. The lever is in balance when the load and block are not on the lever.

The dumpster has a weight W and a center of gravity at G. Determine the force in the hydraulic cylinder needed to hold it in the general position \(\theta\).

The crankshaft is subjected to a torque of \(M = 50\ N \cdot m\). Determine the horizontal compressive force F applied to the piston for equilibrium when \(\theta = 60^{\circ}\).

The crankshaft is subjected to a torque of \(M = 50\ N \cdot m\). Determine the horizontal compressive force F and plot the result of F (ordinate) versus \(\theta\) (abscissa) for \(0^{\circ}\ \leq\ \theta\ \leq\ 90^{\circ}\).

Rods AB and BC have centers of mass located at their midpoints. If all contacting surfaces are smooth and BC has a mass of 100 kg, determine the appropriate mass of AB required for equilibrium.

If the potential energy for a conservative one degree-of-freedom system is expressed by the relation \(V = (4x^3 - x^2 - 3x + 10)\ ft \cdot lb\), where x is given in feet, determine the equilibrium positions and investigate the stability at each position.

If the potential function for a conservative one-degree-of-freedom system is \(V = (8x^3 - 2x^2 - 10)\ J\), where x is given in meters, determine the positions for equilibrium and investigate the stability at each of these positions.

If the potential function for a conservative one-degree-of-freedom system is \(V = (12\ \sin\ 2 \theta + 15\ \cos\ \theta)\ J\), where \(0^{\circ}\ <\ \theta\ <\ 180^{\circ}\), determine the positions for equilibrium and investigate the stability at each of these positions.

The potential energy of a one-degree-of-freedom system is defined by \(V = (20x^3 - 10x^2 - 25x - 10)\ ft \cdot lb\), where x is in ft. Determine the equilibrium positions and investigate the stability for each position.

If the potential function for a conservative one degree-of-freedom system is \(V = (10\ \cos\ 2 \theta + 25\ \sin\ \theta)\ J\), where \(0^{\circ}\ <\ \theta\ <\ 180^{\circ}\), determine the positions for equilibrium and investigate the stability at each of these positions.

Determine the angle \(\theta\) for equilibrium and investigate the stability of the mechanism in this position. The spring has a stiffness of k = 1.5 kN/m and is unstretched when \(\theta = 90^{\circ}\). The block A has a mass of 40 kg. Neglect the mass of the links.

The spring of the scale has an unstretched length of a. Determine the angle \(\theta\) for equilibrium when a weight W is supported on the platform. Neglect the weight of the members. What value W would be required to keep the scale in neutral equilibrium when \(\theta = 0^{\circ}\)?

If the springs at A and C have an unstretched length of 10 in. while the spring at B has an unstretched length of 12 in., determine the height h of the platform when the system is in equilibrium. Investigate the stability of this equilibrium configuration. The package and the platform have a total weight of 150 lb.

The spring is unstretched when \(\theta = 45^{\circ}\) and has a stiffness of k = 1000 lb/ft. Determine the angle \(\theta\) for equilibrium if each of the cylinders weighs 50 lb. Neglect the weight of the members.

The two bars each have a weight of 8 lb. Determine the angle \(\theta\) for the equilibrium and investigate the stability at the equilibrium position. The spring has an unstretched length of 1 ft.

Determine the angle \(\theta\) for equilibrium and investigate the stability at this position. The bars each have a mass of 3 kg and the suspended block D has a mass of 7 kg. Cord DC has a total length of 1 m.

The two bars each have a weight of 8 lb. Determine the required stiffness k of the spring so that the two bars are in equilibrium when \(\theta = 30^{\circ}\). The spring has an unstretched length of 1 ft.

A homogeneous block rests on top of the cylindrical surface. Derive the relationship between the radius of the cylinder, r, and the dimension of the block, b, for stable equilibrium. Hint: Establish the potential energy function for a small angle \(\theta\), i.e., approximate \(\sin\ \theta \approx 0\), and \(\cos\ \theta \approx1-\theta^2/2\).

The assembly shown consists of a semicircular cylinder and a triangular prism. If the prism weighs 8 lb and the cylinder weighs 2 lb, investigate the stability when the assembly is resting in the equilibrium position.

A spring with a torsional stiffness k is attached to the pin at B. It is unstretched when the rod assembly is in the vertical position. Determine the weight W of the block that results in neutral equilibrium. Hint: Establish the potential energy function for a small angle \(\theta\), i.e., approximate \(\sin\ \theta \approx 0\), and \(\cos\ \theta \approx 1-\theta^2/2\).

A spring having a stiffness k = 100 lb/ft and a cylinder weighing 100 lb are attached to the smooth collar. Determine the smallest angle \(\theta\) for equilibrium and investigate the stability at the equilibrium position. The spring has an unstretched length of 2 ft. Neglect the weight of the collar.

The small postal scale consists of a counterweight \(W_1\), connected to the members having negligible weight. Determine the weight \(W_2\) that is on the pan in terms of the angles \(\theta\) and \(\phi\) and the dimensions shown. All members are pin connected.

If the spring has a torsional stiffness \(k = 300\ N \cdot m/rad\) and is unwound when \(\theta = 90^{\circ}\), determine the angle for equilibrium if the sphere has a mass of 20 kg. Investigate the stability at this position. Collar C can slide freely along the vertical guide. Neglect the weight of the rods and collar C.

The truck has a mass of 20 Mg and a mass center at G. Determine the steepest grade \(\theta\) along which it can park without overturning and investigate the stability in this position.

The cylinder is made of two materials such that it has a mass of m and a center of gravity at point G. Show that when G lies above the centroid C of the cylinder, the equilibrium is unstable.

If each of the three links of the mechanism has a weight W, determine the angle \(\theta\) for equilibrium. The spring, which always remains vertical, is unstretched when \(\theta = 0^{\circ}\).

If the uniform rod OA has a mass of 12 kg, determine the mass m that will hold the rod in equilibrium when \(\theta = 30^{\circ}\). Point C is coincident with B when OA is horizontal. Neglect the size of the pulley at B.

The triangular block of weight W rests on the smooth corners which are a distance a apart. If the block has three equal sides of length d, determine the angle \(\theta\) for equilibrium.

Two uniform bars, each having a weight W, are pin connected at their ends. If they are placed over a smooth cylindrical surface, show that the angle \(\theta\) for equilibrium must satisfy the equation \(\cos\ \theta/ \sin^3\ \theta = a/2r\).