Jean, who likes physics experiments, dangles her watch from a thin piece of string while the jetliner she is in takes off from Dulles Airport (Fig. 4-55). She notices that the string makes an angle of 24° with respect to the vertical while the aircraft accelerates for takeoff, which takes about 17 seconds. Estimate the takeoff speed of the aircraft.
To estimate the takeoff speed of the aircraft, we can use the concept of simple harmonic motion. The angle made by the string with respect to the vertical can be considered as the angle of displacement of a simple pendulum.
The time period of a simple pendulum is given by the equation:
T = 2π√(L/g)
where T is the time period, L is the length of the string, and g is the acceleration due to gravity.
Since we want to determine the takeoff speed of the aircraft, we can relate the time period of the pendulum to the time it takes for the aircraft to accelerate for takeoff.
The time period, T, of the pendulum is given by:
T = 2π√(L/a)
where a is the acceleration of the aircraft during takeoff.
Given that the angle is 24° and the time for takeoff acceleration is 17 seconds, we can use the equation:
tan(24°) = a/g
Now, let's rearrange the equation to solve for a:
a = g * tan(24°)
Substituting the value of acceleration due to gravity (approximately 9.8 m/s²) and solving for a, we have:
a ≈ 9.8 m/s² * tan(24°)
Calculating this value, we find:
a ≈ 4.20 m/s²
Now, to estimate the takeoff speed of the aircraft, we can use the equation for acceleration:
a = (v_f - v_i) / t
where a is the acceleration, v_f is the final velocity (takeoff speed), v_i is the initial velocity (0 m/s), and t is the time for takeoff acceleration (17 s).
Rearranging the equation, we have:
v_f = a * t
Substituting the values, we get:
v_f ≈ 4.20 m/s² * 17 s
Calculating this, we find:
v_f ≈ 71.4 m/s
Therefore, the estimated takeoff speed of the aircraft is approximately 71.4 m/s.
To estimate the takeoff speed of the aircraft, we can use the concept of simple harmonic motion and the law of conservation of energy. The angle of the string with respect to the vertical gives us a clue about the acceleration of the aircraft.
Let's break down the problem and explain the steps to find the solution:
Step 1: Understand the problem
We have a jetliner taking off from Dulles Airport. Jean notices that the string, on which her watch is hung, makes an angle of 24° with respect to the vertical while the aircraft accelerates for takeoff. We are required to estimate the takeoff speed of the aircraft.
Step 2: Identify the relevant concepts and formulas
We can apply Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = m*a).
We also need to consider the concept of centripetal force (F = m*v²/r), which is responsible for the circular motion of the watch when it deviates from the vertical.
The tension in the string can be represented as T = m*g*cosθ, where m is the mass of the watch, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the string with respect to the vertical.
Finally, we can use the relationship between acceleration and time for uniformly accelerated motion: a = Δv/Δt, where a is the acceleration, Δv is the change in velocity, and Δt is the time interval (17 seconds in this case).
Step 3: Derive the relevant equations
By substituting the tension equation into the equation for centripetal force and equating it to the net force, we can solve for the velocity of the aircraft.
m*v²/r = m*g*cosθ
v² = g*r*cosθ
Step 4: Calculate the takeoff speed
Now, we can substitute the given values:
θ = 24° = (24π/180) radians (convert degrees to radians)
g = 9.8 m/s² (acceleration due to gravity)
r = length of the string (assumed known or can be measured)
Solving the equation, we find:
v = √(g*r*cosθ)
Step 5: Plug in the values and perform the calculation
Substituting the known values, we can calculate the takeoff speed of the aircraft.
v = √(9.8 * r * cos(24π/180))
By substituting the known values for r, we can obtain the final numerical result.
It's important to remember that the answer will depend on the length of the string, which is not given in the question. Therefore, you will need that information in order to calculate the exact speed.
Meghan, Maddy, Hannah, Cameron -- please do not switch screen names. We know that only one person posted the last 8 physics questions.
You'd have more luck getting help by sticking with one name and showing us what you do and don't know about each problem. Otherwise, it just looks like homework dumping and mooching.