Question

A pendulum has a length L = 1.06 m. It hangs straight down in a jet plane about to take off as shown by the dotted line in the figure.As the jet accelerates uniformly during take-off, the pendulum deflects horizontally by D = 0.420 m to a new equilibrium position. Calculate the magnitude of the plane's acceleration.

Answer 1

Answer: 19.79 m/s²

Step-by-step explanation:

To calculate the magnitude of the plane's acceleration, we can use the concept of centripetal acceleration.

1. The pendulum has a length L = 1.06 m. When the plane accelerates uniformly during take-off, the pendulum deflects horizontally by D = 0.420 m.

2. The deflection of the pendulum is due to the centripetal acceleration acting towards the center of the circular path of the pendulum's motion.

3. The magnitude of the centripetal acceleration can be calculated using the formula:

a = (4π²L) / T²

where a is the centripetal acceleration, L is the length of the pendulum, and T is the period of the pendulum.

4. The period of a pendulum is the time it takes for one complete oscillation. The period can be calculated using the formula:

T = 2π√(L / g)

where g is the acceleration due to gravity (approximately 9.8 m/s²).

5. Substituting the value of L = 1.06 m into the period formula, we get:

T = 2π√(1.06 / 9.8) ≈ 2.126 s

6. Now, we can substitute the values of L and T into the centripetal acceleration formula to calculate the magnitude of the plane's acceleration:

a = (4π²(1.06)) / (2.126)²

a ≈ 19.79 m/s²

Therefore, the magnitude of the plane's acceleration is approximately 19.79 m/s².

I hope this helps :)

Answer 2

The magnitude of the plane's acceleration is determined by analyzing the forces on a deflected pendulum. Using the formula a = g(D/L) and substituting the given values, the acceleration is found to be 3.88 m/s².

Step-by-step explanation:

To calculate the magnitude of the plane's acceleration, we can analyze the forces acting on the pendulum when it is deflected. The deflection occurs due to the horizontal acceleration of the plane, which causes a horizontal force on the pendulum. This force results in a new equilibrium position.

In this new position, two forces act on the pendulum: the tension in the string T, and the weight of the pendulum mg, where m is the mass of the pendulum and g is the acceleration due to gravity. The tension T has two components: one vertical component that balances the weight of the pendulum (Ty = mg), and a horizontal component (Tx) that provides the centripetal force necessary for the circular motion of the pendulum, which is equal to the gravitational force multiplied by the ratio of D to L.

The horizontal component of the tension is responsible for the pendulum's acceleration and can be expressed as Tx = ma, where a is the plane's acceleration. Thus, we have ma = mg(D/L). By cancelling out the m, and rearranging the formula, the plane's acceleration a can be calculated as a = g(D/L).

Given that g = 9.80 m/s², L = 1.06 m, and D = 0.420 m, we can substitute these values into the formula:

a = 9.80 m/s² (0.420 m / 1.06 m)

Upon calculation, we get:

a = 3.88 m/s²

Therefore, the magnitude of the plane's acceleration is 3.88 m/s².