A model airplane of mass 1.5 kg is attached

to a horizontal string and flies in a horizontal
circle of radius 5.8 m, making 2 revolutions
every 8 s. (The weight of the plane is balanced
by the upward “lift” force of the air on the
wings of the plane.)
The accelaration due to the gravity is 9.81
m/s
2
.
Find the speed of the plane.

speed=distance/time=2PI*5.8/4sec m/s

To find the speed of the plane, we can use the formula for centripetal acceleration.

First, let's find the period (T) of one revolution using the information given. The plane completes 2 revolutions in 8 seconds, so the period is calculated as follows:
T = Time / Number of revolutions = 8 s / 2 = 4 s.

Next, we can calculate the angular velocity (ω) using the formula:
ω = 2π / T, where π is pi (approximately 3.14).
ω = 2π / 4 s = π / 2 s^(-1).

Now we can calculate the centripetal acceleration (ac) using the formula:
ac = ω² * r, where r is the radius of the circle.
Since the radius is given as 5.8 m, we have:
ac = (π / 2 s^(-1))² * 5.8 m = (π² / 4) * 5.8 m ≈ 14.57 m/s².

The centripetal acceleration is provided by the net force acting on the plane, which is the difference between the lift force and the weight force.

Since the weight of the plane is balanced by the lift force, we can equate the weight force (mg) to the centripetal force (mac):
mg = mac.

Rearranging the equation to solve for the centripetal force:
F_c = mg / a = (1.5 kg * 9.81 m/s²) / 14.57 m/s² ≈ 1.01 kg*m/s² or N (Newtons).

The centripetal force is also the product of mass and velocity squared divided by the radius:
F_c = m * v² / r.

Rearranging equation to solve for speed (v), we get:
v² = F_c * r / m = 1.01 kg*m/s² * 5.8 m / 1.5 kg = 3.9333 m²/s².

Taking the square root of both sides, we find:
v = √(3.9333 m²/s²) ≈ 1.98 m/s.

Therefore, the speed of the plane is approximately 1.98 m/s.

To find the speed of the plane, we can use the relationship between centripetal acceleration and velocity.

1. First, let's find the angular velocity (ω) of the plane. The plane makes 2 revolutions every 8 seconds, so we need to convert this into radians per second. Since one revolution is equal to 2π radians, the angular velocity can be calculated as follows:
ω = (2 revolutions / 8 seconds) * 2π radians/revolution

2. Now that we have the angular velocity, we can find the centripetal acceleration (ac) using the formula:
ac = ω^2 * radius

3. Since the plane is in rotational motion due to the force of tension in the string, the centripetal acceleration is provided by the tension. The tension force is also equal to the weight of the plane, which is given as the product of the mass (m) and the acceleration due to gravity (g):
ac = Tension / Mass = (m * g) / Mass

4. Equating the two expressions for centripetal acceleration, we have:
(m * g) / Mass = ω^2 * radius

5. Rearranging the equation, we can solve for ω:
ω^2 = (m * g) / (Mass * radius)

6. Now that we have ω, we can calculate the speed (v) using the relationship between linear velocity and angular velocity:
v = ω * radius

Using the given values, we can substitute them into the equations to find the speed of the plane.