The central radius of gyration of the spool is k, and its inner radius is R. If the spool is released from rest, determine the velocity of the spool as a function of its displacement x. Neglect the friction between the spool and the vertical wall.

A person bending forward to lift a load "with his back" rather than "with his knees" can be injured by large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, and to understand why back problems are common among humans, consider the model shown in the figure for a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight $350 \mathrm{~N}$, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is $12.0^{\circ}$. Find the tension in the back muscle and the compressional force in the spine.

When a person stands on tiptoe (a strenuous position), The total gravitational force on the body $\overrightarrow{\mathbf{F}}_g$ is supported by the force $\overrightarrow{\mathbf{n}}$ exerted by the floor on the toes of one foot, where $\overrightarrow{\mathbf{T}}$ is the force exerted by the Achilles tendon on the foot and $\overrightarrow{\mathbf{R}}$ is the force exerted by the tibia on the foot. Find the values of $T, R$, and $\theta$ when $F_g=700 \mathrm{~N}$.

The arm weighs $41.5 \mathrm{~N}$. The gravitational force on the arm acts through point $A$. Determine the magnitudes of the tension force $\overrightarrow{\mathbf{F}}_t$ in the deltoid muscle and the force $\overrightarrow{\mathbf{F}}_s$ exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

Figure shows a claw hammer as it is being used to pull a nail out of a horizontal board. If a force of $150 \mathrm{~N}$ is exerted horizontally as shown, find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume that the force the hammer exerts on the nail is parallel to the nail.

The uniform square steel plate has a mass of $6 \mathrm{~kg}$ and is resting on a smooth horizontal surface in the $x-y$ plane. If a horizontal force $P=120 \mathrm{~N}$ is applied to one corner in the direction shown, determine the magnitude of the initial acceleration of corner $A$.

The uniform slender bar AB(900 mm) has a mass of 8 kg and swings in a vertical plane about the pivot at A. If $\dot{\theta}=2 \operatorname{rad} / \mathrm{s} \text { when } \theta=30^{\circ}$, compute the force supported by the pin at A at that instant.

Determine the angular acceleration of the uniform disk if $(a)$ the rotational inertia of the disk is ignored and $(b)$ the inertia of the disk is considered. The system is released from rest, the cord does not slip on the disk, and bearing friction at $O$ may be neglected.

The 30-in. slender bar weighs 20 lb and is mounted on a vertical shaft at O. If a torque M = 100 lb-in. is applied to the bar through its shaft, calculate the horizontal force R on the bearing as the bar starts to rotate.

The uniform 20-kg slender bar is pivoted at O and swings freely in the vertical plane. If the bar is released from rest in the horizontal position, calculate the initial value of the force R exerted by the bearing on the bar an instant after release.

The uniform slender bar of mass m and length L is released from rest when in the horizontal position. Determine its angular velocity and masscenter speed as it passes the vertical position.

The 15- kg uniform bar AB is in the position shown and moving with velocity $\bar{v}=16$ m / s and angular velocity $\omega=120$ rad / s when end A strikes a rigid obstruction. Determine the velocity of end B just after the impact, assuming that the bar does not rebound.

The uniform 30-kg bar OB is secured to the accelerating frame in the 30$^\circ$ position from the horizontal by the hinge at O and roller at A. If the horizontal acceleration of the frame is a = 20 m/$s^2$, compute the force FA on the roller and the x- and y-components of the force supported by the pin at O.

The 6-kg frame AC and 4-kg uniform slender bar AB of length l slide with negligible friction along the fixed horizontal rod under the action of the 80-N force. Calculate the tension T in wire BC and the x- and y-components of the force exerted on the bar by the pin at A. The x-y plane is vertical.