A 1-lb collar is attached to a spring and slides without friction along a circular rod in a vertical plane. The spring has an undeformed length of 5 in. and a constant k of 25 lb/ft. Knowing that the collar is released from point A, determine the speed of the collar and the normal force between the collar and the rod as the collar passes through point B.

To determine the speed of the collar and the normal force between the collar and the rod as the collar passes through point B, one can use the principles of conservation of mechanical energy and Newton's second law.

Let's denote point A as the point of release and point B as the point where we want to find the speed and normal force.

1. Conservation of Mechanical Energy: Assuming no external work is done (no friction or air resistance), the total mechanical energy is conserved. This means that the sum of the potential energy due to gravity, potential energy stored in the spring (elastic potential energy), and kinetic energy remains constant between points A and B.

Total Mechanical Energy at A = Total Mechanical Energy at B U_gA + U_sA + K_A = U_gB + U_sB + K_B

At the point of release (A), the collar has only potential energy (both gravitational and elastic) and no kinetic energy since it is at rest. - U_gA is the gravitational potential energy at A. - U_sA is the spring potential energy at A. - K_A is the kinetic energy at A, which is 0.

At point B, the collar has kinetic energy, gravitational potential energy, and spring potential energy.

2. Newton's Second Law: Once you have the speed of the collar at point B, you can apply Newton's second law to find the normal force. The force exerted by the spring and the weight components will contribute to the normal force on the circular track.

To apply Newton’s second law, an analysis of the forces in the normal direction to the path at point B is performed. In a vertical plane, the normal force N, at any point, is obtained from the equation of motion in the normal direction, considering the centripetal acceleration (v^2/r).

N + F_s - mgcos(θ) = mv^2/r

Where: - F_s is the force exerted by the spring, - θ is the angle between the vertical and the line from the center of the circle to the collar, - m is the mass of the collar, - v is the speed of the collar at point B, - r is the radius of the circular path at point B, - g is the acceleration due to gravity.

Without specific numbers or a diagram, we can't provide numeric answers, but this is the general approach to solving problems like this.

Extra: The potential energy due to gravity (U_g) is calculated by the formula U_g = mgh, where m is mass, g is the acceleration due to gravity, and h is the vertical height above some reference level. The elastic potential energy stored in a spring (U_s) is given by U_s = 1/2 k x^2, where k is the spring constant, and x is the deformation from the spring's undeformed length. The kinetic energy (K) of a moving object is given by K = 1/2 mv^2, where m is the mass and v is the speed of the object.

For springs, the force exerted by the spring (F_s) is proportional to the displacement from its relaxed length, described by Hooke's Law: F_s = -kx, where k is the spring constant and x is the displacement.

To find the normal force, consider all the forces acting perpendicular to the rod at the point of interest. It's important to separate the forces into components along the axis of interest and use free-body diagrams to help visualize and solve for the unknowns. Normal force often becomes a central focus in problems involving circular motion, as it contributes to the centripetal force necessary to keep the object moving in a curved path.