A toy airplane of mass 0.159 kg is tied to the ceiling with a string. The airplane moves in a horizontal circle of radius 2.97 m, with a constant speed of 1.51 m/s.

What is the angle the string makeswith the vertical?

To find the angle that the string makes with the vertical, we can use the concept of centripetal force and gravitational force.

The centripetal force that keeps the toy airplane moving in a circular path is provided by the tension in the string. The tension in the string is directed toward the center of the circle and is equal to the centripetal force.

The gravitational force on the toy airplane is directed vertically downward and is equal to the weight of the airplane, which is the mass of the airplane multiplied by the acceleration due to gravity.

Since the toy airplane is in equilibrium, the vertical component of the tension in the string must equal the gravitational force. This vertical component of tension counteracts the weight of the airplane.

Thus, we can set up the equation:

T * cos(θ) = mg

Where T is the tension in the string, θ is the angle the string makes with the vertical, m is the mass of the airplane, and g is the acceleration due to gravity.

Solving for θ, we have:

cos(θ) = mg / T

Now, we need to find the tension in the string. The tension can be found using the centripetal force:

T = mv^2 / r

Where v is the constant speed of the airplane and r is the radius of the circular path.

Substituting this into the equation for cos(θ), we have:

cos(θ) = mg / (mv^2 / r)
cos(θ) = rg / v^2

Now, we can substitute the given values into the equation to calculate the angle:

cos(θ) = (2.97 m)(9.8 m/s^2) / (1.51 m/s)^2

Solving for cos(θ) using a calculator, we have:

cos(θ) ≈ 0.577

Finally, we can find the angle θ by taking the inverse cosine of 0.577:

θ ≈ cos^(-1)(0.577)

Calculating this using a calculator, we find:

θ ≈ 55.4 degrees

Therefore, the angle the string makes with the vertical is approximately 55.4 degrees.

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