The image you uploaded contains multiple physics problems involving forces, kinematics, and dynamics. I'll go through each one and summarize the steps in a simpler way. Here’s a step-by-step guide for each problem:

Problem 13-64

• Given: A jet plane traveling at a speed of 1000 ft/s along a parabolic curve $y = 20(10^{-6})x^2 + 5000$. The pilot weighs 180 lb.
• Find: Normal and tangential components of the force the seat exerts when $x = 1000$.

Steps:

1. Find the slope (dy/dx) of the curve to get the tangential direction.
2. Calculate the radius of curvature ($\rho$) using second derivatives.
3. Use $F_n = m v^2/\rho$ for the normal force, and $F_t = m a_t$ for tangential force.

Problem 13-63

• Given: A plane traveling at 800 ft/s, along a parabolic curve $y = 20(10^{-6})x^2 + 5000$, with pilot weight of 180 lb.
• Find: Normal and tangential forces when at the lowest point.

Steps:

1. Use the same curve formula as Problem 13-64.
2. The lowest point means the velocity is purely horizontal, so the tangential force is zero.
3. Use the radius of curvature and calculate the normal force.

Problem 13-14

• Given: A truck brakes at 15 m/s and comes to a stop over a distance of 10 m, with a trailer attached.
• Find: Force developed in the coupling.

Steps:

1. Find deceleration (a) using kinematic equations.
2. Apply Newton’s second law for the truck and trailer to get the force.

Problem 13-15

• Given: A motor lifts a crate (90 kg) with an acceleration of 6 m/s².
• Find: Force reactions and moments at the fixed support.

Steps:

1. Calculate tension (T) in the cable using $T = m(a + g)$.
2. Resolve forces at the support using equilibrium equations.

Problem 13-38 (Pendulum)

• Given: A pendulum bob moves with a velocity of 5 m/s when at $\theta = 30^\circ$.
• Find: Initial tension in the cord.

Steps:

1. Use energy conservation to find speed and relate to tension using $T - mg \cos \theta = m v^2 / r$.

Problem 13-41 (Pendulum)

• Given: A pendulum bob at a 60° angle with a velocity of 8 m/s.
• Find: Tension in the cord and angle of the velocity vector.

Steps:

1. Use similar energy conservation and free-body diagram as Problem 13-38 to solve for tension.

Would you like me to provide further details or clarifications for any specific part of these problems?

Here are some follow-up questions you might explore:

1. How does the radius of curvature relate to the normal force in circular motion?
2. Why is the tangential force zero at the lowest point of the curve for the pilot?
3. How does the tension in a pendulum cord vary with the angle?
4. What role does deceleration play in the force developed in the truck-trailer system?
5. How would the solution change if the acceleration of the motor lifting the crate was different?

Tip: When solving dynamics problems, always start by identifying forces acting on the object and use free-body diagrams to guide the solution.