A 8.0×10−2-{\rm kg} toy airplane is tied to the ceiling with a string. When the airplane's motor is started, it moves with a constant speed of 1.30 m/s in a horizontal circle of radius 0.41 m,

Find the angle the string makes with the vertical.

To find the angle the string makes with the vertical, we can use the concept of centripetal force acting on the toy airplane.

The centripetal force is provided by the tension in the string, and since the airplane is moving in a horizontal circle, the centripetal force is directed towards the center of the circle.

We can start by calculating the centripetal force using the mass of the airplane and its speed:

F = ma

where F is the centripetal force, m is the mass of the airplane, and a is the centripetal acceleration.

The centripetal acceleration is given by:

a = v^2 / r

where v is the speed of the airplane and r is the radius of the circle.

Plugging in the given values:

m = 8.0×10^-2 kg
v = 1.30 m/s
r = 0.41 m

a = (1.30 m/s)^2 / 0.41 m
a = 4.03 m^2/s^2

Now, we can calculate the centripetal force:

F = (8.0×10^-2 kg) * (4.03 m^2/s^2)
F = 0.0322 N

The tension in the string provides this centripetal force. Let's assume the angle the string makes with the vertical is θ.

Therefore, the vertical component of the tension cancels out the weight of the airplane, and the horizontal component provides the centripetal force.

The vertical component of the tension is given by:

T * cos(θ) = mg

where T is the tension and g is the acceleration due to gravity.

The horizontal component of the tension is the centripetal force:

T * sin(θ) = F

We can divide these two equations to eliminate T:

(T * sin(θ)) / (T * cos(θ)) = F / mg

tan(θ) = F / mg

Now we can substitute the values and solve for θ:

tan(θ) = (0.0322 N) / (8.0×10^-2 kg * 9.8 m/s^2)
tan(θ) = 0.041

Taking the inverse tangent of both sides:

θ = arctan(0.041)
θ ≈ 2.36 degrees

Therefore, the angle the string makes with the vertical is approximately 2.36 degrees.

To find the angle the string makes with the vertical, we need to analyze the forces acting on the toy airplane.

When the toy airplane is moving in a horizontal circle, it experiences a centripetal force directed towards the center of the circle. In this case, the centripetal force is provided by the tension in the string.

Let's analyze the forces acting on the toy airplane:

1. Weight (mg): This force acts vertically downwards and is equal to the mass of the toy airplane (m) multiplied by the acceleration due to gravity (g).

2. Tension in the string (T): This force acts along the string and provides the centripetal force required to keep the toy airplane moving in a circle.

Now, we can set up the equation for the forces in the vertical direction:

T * cos(θ) - mg = 0

Here, θ is the angle the string makes with the vertical. The component of tension along the vertical direction is T * cos(θ).

Since the toy airplane is moving with a constant speed, the net force acting on it must be zero. Hence, we can set up the equation for the forces in the horizontal direction:

T * sin(θ) = mv^2 / r

Here, v is the speed of the toy airplane (1.30 m/s) and r is the radius of the circle (0.41 m).

We can now solve these two equations simultaneously to find the angle θ.

First, rearrange the equation for T * cos(θ) - mg = 0 to solve for T:

T * cos(θ) = mg

Next, substitute the value of T in the equation T * sin(θ) = mv^2 / r:

mg * sin(θ) = mv^2 / r

Rearrange the equation to solve for sin(θ):

sin(θ) = (v^2 / r) / g

Now, substitute the given values into the equation and calculate sin(θ):

sin(θ) = (1.30^2 / 0.41) / 9.8

sin(θ) ≈ 0.5537

Finally, find the angle θ by taking the inverse sine (sin^(-1)) of the result:

θ ≈ sin^(-1)(0.5537)

θ ≈ 33.85 degrees

Therefore, the string makes an angle of approximately 33.85 degrees with the vertical.