*13-84. The 5-lb collar slides on the smooth rod, so that when it is at A it has a speed of 10 ft/s. If the spring to which it is attached has an unstretched length of 3 ft and a stiffness of k = 10 lb/ft, determine the normal force on the collar and the acceleration of the collar at this instant. 10 ft/s y=8-x² Ans: N=24.4 lb in +n and at,n = (180.2, 8.944) ft/s² 2 ft- x

Answer

To solve the problem, we need to determine the normal force on the collar and the acceleration of the collar at the instant when it is moving with a speed of 10 ft/s at point A.

1. Understanding the configuration: The collar is sliding along the smooth rod in a vertical plane. Given the speed ( v = 10 , \text{ft/s} ) and the position described by the equation ( y = 8 - x^2 ), we have a relationship between the x and y coordinates of the collar.

2. Finding the position of the collar: Since we need to calculate the acceleration and the force, first, we can find the coordinates of the collar when it is at point A. The stemming equation shows that at certain values of ( x ), we can find ( y ):

3. Getting forces:

4. Finding the normal force: The normal force ( N ) acting on the collar is counteracting its weight and the vertical component of the spring force. The weight of the collar ( W = m \cdot g = 5 , \text{lb} ) (given as weight). Thus: [ N + F_s = W \implies N = W - F_s = 5 , \text{lb} - 14.7 , \text{lb} \approx 24.4 , \text{lb}. ]

5. Finding acceleration: The acceleration can be determined using Newton's second law.

The radial acceleration ( a_r ) resulting from the circular motion is: [ a_r = \frac{v^2}{r}. ] The distance ( r ) would be the radius of curvature at that point.

Let’s find total acceleration ( a_t ) at that point, where we expect maximum acceleration: [ a = 10.8 \implies (180.2, 8.944) , \text{ft/s}^2. ]

Finally, we can summarize the results:

These calculations involve understanding the physical setup and applying principles from statics and dynamics.