You plan to fly a model airplane of mass 0.655 kg that is attached to a horizontal string. The plane will travel in a horizontal circle of radius 5.50 m. (Assume the weight of the plane is balanced by the upward "lift" force of the air on the wings of the plane.) The plane will make 1.15 revolutions every 4.50 s.
(a) Find the speed at which you must fly the plane.
m/s
(b) Find the force exerted on your hand as you hold the string (assume the string is massless).
N
To find the speed at which you must fly the plane, you can start by calculating the angular velocity.
(a) Angular velocity (ω) can be calculated using the formula:
ω = 2π * (number of revolutions / time)
Plugging in the given values, we have:
ω = 2π * (1.15 rev / 4.50 s)
Now, let's calculate the angular velocity:
ω = 2π * (1.15 rev / 4.50 s)
ω ≈ 4.038 rad/s
The centripetal speed (v) of the plane can be found using the formula:
v = ω * r
where r is the radius of the circle.
(b) Now, let's find the speed (v) of the plane:
v = ω * r
v = 4.038 rad/s * 5.50 m
v ≈ 22.21 m/s
To find the force exerted on your hand as you hold the string, you need to consider the tension force in the string. This force provides the centripetal force required to keep the plane moving in a circle.
The centripetal force (F) can be calculated using the formula:
F = (m * v^2) / r
where m is the mass of the plane, and v is its speed.
Now, let's calculate the force exerted on your hand:
F = (m * v^2) / r
F = (0.655 kg * (22.21 m/s)^2) / 5.50 m
F ≈ 11.53 N
Therefore, the force exerted on your hand as you hold the string is approximately 11.53 N.
To find the speed at which you must fly the plane, you can use the formula for the speed of an object moving in a circle:
v = (2πr) / T
Where:
v = speed
π = pi (approximately 3.14159)
r = radius of the circle
T = time period for one revolution
In this case, the radius of the circle is given as 5.50 m and the time period for one revolution is given as 4.50 s. Plug these values into the formula to calculate the speed:
v = (2π * 5.50) / 4.50
Now, let's solve it:
v = 34.557 m/s
So, the speed at which you must fly the plane is approximately 34.557 m/s.
Now, let's move on to part (b) to find the force exerted on your hand as you hold the string.
The force exerted on the string is the centripetal force required to keep the plane moving in a circle. The formula for centripetal force is:
F = (m v^2) / r
Where:
F = force exerted on the string
m = mass of the plane
v = speed of the plane
r = radius of the circle
In this case, the mass of the plane is given as 0.655 kg and the speed of the plane is 34.557 m/s. The radius of the circle is still 5.50 m. Plug these values into the formula to calculate the force:
F = (0.655 * 34.557^2) / 5.50
Now, let's solve it:
F = 17.431 N
So, the force exerted on your hand as you hold the string is approximately 17.431 N.