The uniform bar of mass m and length L is supported in the vertical plane by two

The potential energy of a mechanical system is given by \(V=6 x^3-9 x^2+7\), where x is the position coordinate associated with its single degree of freedom. Determine the position or positions of equilibrium of the system and the stability condition of the system at each equilibrium position.

The torsional spring at A has a stiffness \(k_T\) and is undeformed when bars OA and AB are in the vertical position and overlap. Each uniform bar has mass m. Determine the equilibrium configurations of the system over the range \(0 \leq \theta \leq 90^{\circ}\) and the stability of the system at each equilibrium position for m = 1.25 kg, b = 750 mm, and \(k_T=1.8 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}\).

The uniform bar of mass m and length L is supported in the vertical plane by two identical springs each of stiffness k and compressed a distance in the vertical position \(\theta\) = 0. Determine the minimum stiffness k which will ensure a stable equilibrium position with \(\theta\) = 0. The springs may be assumed to act in the horizontal direction during small angular motion of the bar.

The uniform partial circular ring of radius r and mass m is pivoted about O by the light support link OA shown. Determine the maximum value of h for which equilibrium in the position shown is stable. Evaluate for \((a) \theta=30^{\circ},(b) \theta=45^{\circ},(c) \theta=60^{\circ}\), and (d) \(\theta=90^{\circ}\).

Determine the force P required to maintain equilibrium of the spring-loaded mechanism for a given angle \(\theta\). The spring has a stiffness k and is uncompressed at \(\theta=0\). The mass of the parts may be neglected.

In the mechanism shown, the rod AB slides through the pivoted collar at O and compresses the spring when the couple M is applied to link GF to increase the angle \(\theta\). The spring has a stiffness k and would be uncompressed in the position \(\theta\) = 0. Determine the angle \(\theta\) for equilibrium. The weights of the links are negligible. Figure CDEF is a parallelogram.

The cylinder of mass M and radius R rolls without slipping on the circular surface of radius 3R. Attached to the cylinder is a small body of mass m. Determine the required relationship between M and m if the body is to be stable in the equilibrium position shown.

The figure shows the cross section of a uniform 60-kg ventilator door hinged along its upper horizontal edge at O. The door is controlled by the spring-loaded cable which passes over the small pulley at A. The spring has a stiffness of 160 N per meter of stretch and is undeformed when \(\theta\) = 0. Determine the angle for equilibrium.

The body consisting of the solid hemisphere (radius r and density \(\rho_1\)) and concentric right-circular cone (base radius r, height h, and density \(\rho_2\)) is resting on a horizontal surface. Determine the maximum height h which the cone may have without causing the body to be unstable in the upright position shown. Evaluate for the case where (a) the hemisphere and cone are made of the same material, (b) the hemisphere is made of steel and the cone is made of aluminum, and (c) the hemisphere is made of aluminum and the cone is made of steel.

Each of the two gears carries an eccentric mass m and is free to rotate in the vertical plane about its bearing. Determine the values of \(\theta\) for equilibrium and identify the type of equilibrium for each position.

One of the critical requirements in the design of an artificial leg for an amputee is to prevent the knee joint from buckling under load when the leg is straight. As a first approximation, simulate the artificial leg by the two light links with a torsion spring at their common joint. The spring develops a torque M = \(k_T\)\(\beta\), which is proportional to the angle of bend \(\beta\) at the joint. Determine the minimum value of \(k_T\) which will ensure stability of the knee joint for \(\beta\) = 0.

The handle is fastened to one of the spring-connected gears, which are mounted in fixed bearings. The spring of stiffness k connects two pins mounted in the faces of the gears. When the handle is in the vertical position, \(\theta\) = 0 and the spring force is zero. Determine the force P required to maintain equilibrium at an angle \(\theta\).

Determine the maximum height h of the mass m for which the inverted pendulum will be stable in the vertical position shown. Each of the springs has a stiffness k, and they have equal precompressions in this position. Neglect the mass of the remainder of the mechanism.

One end of the torsion spring is secured to the ground at A, and the other end is fastened to the shaft at B. The torsional stiffness \(k_T\) of the elastic spring is the torque required to twist the spring through an angle of one radian. The spring resists the moment about the shaft axis caused by the tension mg in the cable wrapped around the drum of radius r. Determine the equilibrium value of h measured from the dashed position, where the spring is untwisted.

In the figure is shown a small industrial lift with a foot release. There are four identical springs, two on each side of the central shaft. The stiffness of each pair of springs is 2k. In designing the lift, specify the value of k which will ensure stable equilibrium when the lift supports a load (weight) L in the position where \(\theta\) = 0 with no force P on the pedal. The springs are equally precompressed initially and may be assumed to act in the horizontal direction at all times.

The two uniform links, each of mass m, lie in a vertical plane and are connected and constrained as shown. Rod AB is connected to the roller at B and passes through a pivoted collar at A. At the position \(\theta\) = \(\theta_0\), the stop C rests against roller A and the spring is uncompressed. As force P is applied perpendicular to link AE, the angle \(\theta\) increases and the spring of stiffness k is compressed. Determine the force P which will produce equilibrium at an arbitrary angle \(\theta\) > \(\theta_0\).

The uniform link AB has a mass m, and its left end A travels freely in the fixed horizontal slot. End B is attached to the vertical plunger, which compresses the spring as B falls. The spring would be uncompressed at the position \(\theta\) = 0. Determine the angle for equilibrium (other than the impossible position corresponding to \(\theta\) = \(90^{\circ}\)) and designate the condition which will ensure stability.

The 3-lb pendulum swings about axis O-O and has a mass center at G. When \(\theta\) = 0, each spring has an initial stretch of 4 in. Calculate the maximum stiffness k of each of the parallel springs which will allow the pendulum to be in stable equilibrium at the bottom position \(\theta\) = 0.

Predict through calculation whether the homogeneous semicylinder and the half-cylindrical shell will remain in the positions shown or whether they will roll off the lower cylinders.

The figure shows a tilting desk chair together with the design detail of the spring-loaded tilting mechanism. The frame of the seat is pivoted about the fixed point O on the base. The increase in distance between A and B as the chair tilts back about O is the increase in compression of the spring. The spring, which has a stiffness of 96 kN/m, is uncompressed when \(\theta\) = 0. For small angles of tilt it may be assumed with negligible error that the axis of the spring remains parallel to the seat. The center of mass of an 80-kg person who sits in the chair is at G on a line through O perpendicular to the seat. Determine the angle of tilt \(\theta\) for equilibrium. (Hint: The deformation of the spring may be visualized by allowing the base to tilt through the required angle \(\theta\) about O while the seat is held in a fixed position.

The double-axle front suspension is used on small trucks. In a test of the designed action, the frame F must be jacked up so that h = 350 mm in order to relieve the compression in the coil springs. Determine the value of h when the jack is removed. Each spring has a stiffness of 120 kN/m. The load L is 12 kN, and the central frame F has a mass of 40 kg. Each wheel and attached link has a mass of 35 kg with a center of mass 680 mm from the vertical centerline.