A fishing line will break when the tension in it reaches 15 N. A
3.1 m length of it is used to tie a model aeroplane of mass 280
g to a post so it goes round in circles. What is the fastest speed
the aeroplane can reach before the line breaks.
Give your answer both as an angular velocity and in m/s
m * v^2 / r = 15 N
.28 kg * v^2 = 15 N * 3.1 m
Well, let's see here. To find the fastest speed the aeroplane can reach before the line breaks, we need to consider the tension in the line. Now, tension is related to speed through the centripetal force, which is given by the equation F = m*v^2/r, where F is the force, m is the mass, v is the velocity, and r is the radius of the circular motion.
Given that the tension in the line is 15 N, we can set up the equation:
15 N = (0.280 kg)*(v^2)/(3.1 m).
Now, let's solve for v^2:
v^2 = (15 N)*(3.1 m) / (0.280 kg).
v^2 ≈ 165.53 m^2/s^2.
To find the fastest speed, we take the square root of v^2:
v ≈ √(165.53) ≈ 12.86 m/s.
But since we also want the answer in angular velocity, we need to convert this linear speed to angular velocity by dividing it by the radius of the circular motion:
ω = v/r = 12.86 m/s / 3.1 m ≈ 4.15 rad/s.
So, the fastest speed the aeroplane can reach before the line breaks is approximately 12.86 m/s or 4.15 rad/s. Just be careful not to tie any knots in the line like a clown ties balloon animals - that would be a real "knot-ical" disaster!
To find the maximum speed the aeroplane can reach before the fishing line breaks, we need to calculate the tension in the fishing line when the aeroplane is at its maximum speed.
1. Determine the tension in the fishing line:
Since the aeroplane is going around in circles, we can use the centripetal force equation:
Tension = Mass * Centripetal acceleration
The mass of the aeroplane is given as 280 g = 0.28 kg.
The centripetal acceleration can be calculated using the formula:
Centripetal acceleration = (velocity^2) / (radius)
To find the radius, we need to convert the length of the fishing line to a radius by dividing it by 2π:
Radius = Length / (2π)
Radius = 3.1 m / (2π) ≈ 0.4938 m
2. Calculate the centripetal acceleration:
Centripetal acceleration = (velocity^2) / (radius)
Since we are looking for the maximum speed, the centripetal acceleration is at its maximum, which occurs at this point.
3. Substitute the known values into the equation:
Tension = Mass * Centripetal acceleration
15 N = 0.28 kg * (velocity^2) / 0.4938 m
4. Solve for the angular velocity:
Using the formula for angular velocity, we can find the maximum speed in terms of angular velocity:
angular velocity = velocity / radius
velocity = angular velocity * radius
Substituting this into the equation, we get:
15 N = 0.28 kg * (angular velocity * 0.4938 m)^2 / 0.4938 m
Simplifying the equation:
15 N = 0.28 kg * angular velocity^2
Solving for angular velocity:
angular velocity^2 = 15 N / (0.28 kg)
angular velocity^2 ≈ 53.571
angular velocity ≈ √53.571
angular velocity ≈ 7.315 rad/s (rounded to three decimal places)
5. Convert angular velocity to m/s:
To convert angular velocity to m/s, we need to multiply it by the radius.
velocity = angular velocity * radius
velocity = 7.315 rad/s * 0.4938 m
velocity ≈ 3.611 m/s (rounded to three decimal places)
Therefore, the fastest speed the aeroplane can reach before the line breaks is approximately 7.315 rad/s (angular velocity) and 3.611 m/s (meters per second).
To find the fastest speed the aeroplane can reach before the fishing line breaks, we need to consider the tension in the line and the centripetal force acting on the aeroplane.
First, let's convert the mass of the aeroplane from grams to kilograms:
280 g = 0.280 kg
Next, let's calculate the centripetal force required to keep the aeroplane moving in a circle. The centripetal force (Fc) is given by the equation:
Fc = m * v^2 / r
where:
m = mass of the aeroplane (0.280 kg)
v = velocity of the aeroplane (unknown)
r = radius of the circular path (3.1 m)
Now, we can rearrange the equation to solve for v:
v^2 = (Fc * r) / m
To find the maximum velocity, we need to calculate the maximum centripetal force that the fishing line can withstand before breaking. In this case, it is given as 15 N.
Let's substitute the values into the equation:
v^2 = (15 N * 3.1 m) / 0.280 kg
v^2 = 165.71 m^2/s^2
Taking the square root of both sides, we get:
v = √(165.71 m^2/s^2)
v ≈ 12.88 m/s
So, the fastest speed the aeroplane can reach before the line breaks is approximately 12.88 m/s.
Now, to express the answer in terms of angular velocity, we can calculate it using the formula:
ω = v / r
where:
ω = angular velocity
v = velocity (12.88 m/s)
r = radius (3.1 m)
Substituting the values, we have:
ω = (12.88 m/s) / (3.1 m)
ω ≈ 4.16 rad/s
Therefore, the answer in terms of angular velocity is approximately 4.16 rad/s and in m/s is approximately 12.88 m/s.