Solved: The spring of the scale has an unstretched length

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

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

If a force of P = 5 lb is applied to the handle of the mechanism, determine the force the screw exerts on the cork of the bottle. The screw is attached to the pin at A and passes through the collar that is attached to the bottle neck at B.

The disk has a weight of 10 lb and is subjected to a vertical force P = 8 lb and a couple moment M = 8 lb \(\cdot\) ft. Determine the disk’s rotation \(\theta\) if the end of the spring wraps around the periphery of the disk as the disk turns. The spring is originally unstretched.

The punch press consists of the ram R, connecting rod AB, and a flywheel. If a torque of M = 75 N \(\cdot\) m is applied to the flywheel, determine the force F applied at the ram to hold the rod in the position \(\theta=^{\circ}\)

The flywheel is subjected to a torque of M = 75 N \(\cdot\) m. Determine the horizontal compressive force F and plot the result of F (ordinate) versus the equilibrium position \(\theta\) (abscissa) for \(0^{\circ} \leq \theta \leq 180^{\circ}\).

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 bar is supported by the spring and smooth collar that allows the spring to be always perpendicular to the bar for any angle \(\theta\). If the unstretched length of the spring is \(\bar{I}_{0}\), determine the force P needed to hold the bar in the equilibrium position \(\theta\). Neglect the weight of the bar.

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.

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

If each of the three links of the mechanism have a mass of 4 kg, determine the angle \(\theta\) for equilibrium. The spring, which always remains vertical, is unstretched when \(\theta=0^{\circ}\)

The disk is subjected to a couple moment M. Determine the disk’s rotation \(\theta\) required for equilibrium. The end of the spring wraps around the periphery of the disk as the disk turns. The spring is originally unstretched

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 = 200N/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.

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 ends, 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 \(\theta\).

The members of the mechanism are pin connected. If a vertical force of 800 N acts at A, determine the angle \(\theta\) for equilibrium. The spring is unstretched when \(\theta=0^{\circ}\). Neglect the mass of the links.

When \(\theta=30^{\circ}\), the 25-kg uniform block compresses the two horizontal springs 100 mm. Determine the magnitude of the applied couple moments M needed to maintain equilibrium. Take k = 3 kN/m and neglect the mass of the links.

The “Nuremberg scissors” is subjected to a horizontal force of P = 600 N. Determine the angle \(\theta\) for equilibrium. The spring has a stiffness of k = 15 kN/m and is unstretched when \(\theta=15^{\circ}\).

The “Nuremberg scissors” is subjected to a horizontal force of P = 600 N. Determine the stiffness k of the spring for equilibrium when \(\theta=60^{\circ}\). The spring is unstretched when \(\theta=15^{\circ}\).

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 \(0^{\circ} \leq \theta \leq 90^{\circ}\)

The spring is unstretched when \(\theta=0^{\circ}\). If P = 8 lb, determine the angle \(\theta\) for equilibrium. Due to the roller guide, the spring always remains vertical. Neglect the weight 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 potential energy of a one-degree-of-freedom system is defined by \(V=\left(20 x^{3}-10 x^{2}-25 x-10\right) \mathrm{ft} \cdot \mathrm{lb}\), where x is in ft. Determine the equilibrium positions and investigate the stability for each position.

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

If the potential function for a conservative onedegree-of-freedom system is \(V=\left(8 x^{3}-2 x^{2}-10\right) \mathrm{J}\), 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 onedegree-of-freedom system is \(V=(10 \cos 2 \theta+25 \sin \theta) \mathrm{J}\), where \(0^{\circ}<\theta<180^{\circ}\), determine the positions for equilibrium and investigate the stability at each of these positions.

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

The uniform link AB, has a mass of 3 kg and is pin connected at both of its ends. The rod BD, having negligible weight, passes through a swivel block at C. If the spring has a stiffness of k = 100N/m and is unstretched when \(\theta=0^{\circ}\), determine the angle \(\theta\) for equilibrium and investigate the stability at the equilibrium position. Neglect the size of the swivel block

The spring of the scale has an unstretched length of a. Determine the angle \(\theta\)u 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 \(\theta=0^{\circ}\)?

The uniform bar has a mass of 80 kg. Determine the angle \(\theta\) for equilibrium and investigate the stability of the bar when it is in this position. The spring has an unstretched length when \(\theta=90^{\circ}\)

The uniform bar AD has a mass of 20 kg. If the attached spring is unstretched when \(\theta=90^{\circ}\), determine the angle \(\theta\) for equilibrium. Note that the spring always remains in the vertical position due to the roller guide. Investigate the stability of the bar when it is in the equilibrium position.

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

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.

Determine the angle \(\theta\) for equilibrium and investigate the stability at this position. The bars each have a mass of 10 kg and the spring has an unstretched length of 100 mm.

The two bars each have a mass of 8 kg. Determine the required stiffness k of the spring so that the two bars are in equilibrium when \(\theta=60^{\circ}\). The spring has an unstretched length of 1 m. Investigate the stability of the system at the equilibrium position.

A spring with a torsional stiffness k is attached to the hinge 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 \(\sin \theta \approx 0 \text {, and } \cos \theta \approx 1-\theta^{2} / 2 \text {. }\).

A conical hole is drilled into the bottom of the cylinder, which is supported on the fulcrum at A. Determine the minimum distance d in order for it to remain in stable equilibrium.

The uniform rod has a mass of 100 kg. If the spring is unstretched when \(\theta=60^{\circ}\), determine the angle \(\theta\) for equilibrium and investigate the stability at the equilibrium position. The spring is always in the horizontal position due to the roller guide at B.

Each bar has a mass per length of \(m_{0}\). Determine the angles \(\theta\) and \(\phi\) at which they are suspended in equilibrium. The contact at A is smooth, and both are pin connected at B.

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 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

A 3-lb weight is attached to the end of rod ABC. If the rod is supported by a smooth slider block at C and rod BD, determine the angle \(\theta\) for equilibrium. Neglect the weight of the rods and the slider.

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 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

The bent rod has a weight of 5 lb/ft. A pivot is attached at its center A and the rod is balanced as shown. Determine the length L of its vertical segments so that it remains in neutral equilibrium. Neglect the thickness of the rod.

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.