Question

A 5.4-kg bowling ball is attached to the end of a nylon (Young's modulus 3.7x10^9 N/m^2) cord with a cross- sectional area of 3.7x10^5 m^2. The other end of the cord is fixed to the ceiling. When the bowling ball is pulled to one side and released from rest, it swings downward in a circular arc. At the instant it reaches its lowest point, the bowling ball is 1.1 m lower than the point it was released from, and the cord is stretched 3.5x10^-3 m from its unstrained length. What is the unstrained length of the cord? Hint: When calculating any quantity other than the strain, ignore the increase in length of the cord.

Answer 1

The unstrained length of the nylon cord can be found using Young's modulus, the force exerted by the weight of the bowling ball, and by ignoring the additional elongation during the calculation as the problem suggests; thus L is determined using the formula derived from Young's modulus.

Step-by-step explanation:

To find the unstrained length of the nylon cord, we'll need to understand the relationship between the extension (ΔL), Young's modulus (Y), the cross-sectional area (A), and the force applied (F). The stress (F/A) is given by the force exerted by the weight of the bowling ball (mass × gravitational acceleration), and strain (ΔL divided by the original length L) is the relative change in length. By the problem's hint, we know that we can ignore the change in cord length when calculating the stretch, so we'll use the vertical distance the ball fell (1.1 m) as the effective length increase due to stretching (ignoring this additional 3.5 ×10⁻³ m).

First, calculate the force exerted by the bowling ball (F = mass × gravity):

• F = 5.4 kg × 9.8 m/s²

Next, use the relationship for Young's modulus:

• Y = (F/A) / (ΔL/L)

Solve for L:

• L = F / (Y × A × ΔL)

Finally, since we ignore the change in length:

• ΔL = 1.1 m

Insert the values we have to find the unstrained length:

• L = (5.4 kg × 9.8 m/s²) / (3.7×10⁹ N/m² × 3.7×10⁵ m² × 1.1 m)

With this approach, we assume the given change in length (3.5 ×10⁻³ m) is solely due to the weight of the bowling ball, thus giving us the original unstrained length L.

Answer 2

The unstrained length of the nylon cord is found by subtracting the stretch caused by the weight of the bowling ball from the total length of the cord when the ball is at its lowest point. The unstrained length is calculated to be 1.0965 meters.

Step-by-step explanation:

To find the unstrained length of the cord, we must first understand that at the lowest point of the swing, the only forces acting on the bowling ball are the tension in the cord and gravity. Since the increase in the length of the cord due to stretching is to be ignored when calculating any quantity other than strain, we can initially treat the cord as though its length is constant to find the unstrained length.

The ball drops a vertical distance of 1.1 meters from the point of release to the lowest point of its swing. This difference in height corresponds to the total length of the cord plus the displacement due to the stretching, minus the radius of the circular arc traced by the bowling ball during its swing, which also corresponds to the length of the cord.

Thus, if L is the original unstrained length of the cord and ΔL is the stretch (3.5x10^-3 m), the total length of the cord when the ball is at its lowest point would be L + ΔL. The radius of the circular arc is L, hence the vertical distance of 1.1 meters is L + ΔL - L = ΔL. We can solve this for L by rearranging the equation to L = vertical distance - ΔL.

L = 1.1 m - 3.5x10^-3 m = 1.0965 m unstrained length of the cord.