A child finds a rope hanging vertically from the ceiling of a large storage hangar. The child grabs the rope and starts running in a circle. The length of the rope is 13.0m. When the child runs in a circle of radius 7.0m, the child is about to lose contact with the floor. How fast is the child running at that time?
angle of rope from vertical = A
R = 7
T cos A = m g
T sin A = m v^2/R
so
tan A = v^2/(g R) = v^2/(9.81*7)
but we are given that sin A = 7/13 so A = sin^-1 (.538) = 32.6 deg
so
tan A = .639
so
.639 = v^2/(9.81*7)
Well, I have to say, that child must be quite a circus performer! Running in circles while holding onto a rope, that's some serious talent. But let's get back to the question at hand.
To figure out how fast the child is running at the moment they're about to lose contact with the floor, we need to use a little bit of math.
First, we can use the Pythagorean theorem to find the height of the rope above the floor. Since the radius of the circle is 7.0m and the length of the rope is 13.0m, we can calculate the height as the square root of (13.0^2 - 7.0^2).
Once we know the height, we can use some trigonometry to find the distance the child has run along the circumference of the circle. This distance is the same as the angle (in radians) that the child has traveled around the circle multiplied by the radius of the circle.
Now, to find the speed, we can divide the distance traveled (the arc length) by the time it took to travel that distance. But the question doesn't provide any information about time, so unfortunately, we can't determine the speed.
Looks like this circus performance is going to remain a mystery!
To find the speed at which the child is running when they are about to lose contact with the floor, we can use the concept of centripetal force.
The centripetal force acting on the child is provided by the tension in the rope. At the point where the child is about to lose contact with the floor, the tension in the rope must be equal to the weight of the child.
The tension in the rope can be calculated using the formula:
Tension = (mass * velocity^2) / radius
The weight of the child can be calculated using the formula:
Weight = mass * gravitational acceleration
Since we know the radius of the circle (7.0 m) and the length of the rope (13.0 m), we can calculate the velocity at which the child is running using the formula for the circumference of a circle:
Circumference = 2 * π * radius
Let's proceed step by step to solve this problem:
Step 1: Calculate the circumference of the circle
Circumference = 2 * π * 7.0 m
Circumference ≈ 43.98 m
Step 2: Set up the equation for the tension in the rope
Tension = (mass * velocity^2) / radius
Step 3: Replace the tension with the weight of the child
Weight = mass * gravitational acceleration
Step 4: Equate the tension and weight
(mass * velocity^2) / radius = mass * gravitational acceleration
Step 5: Cancel out the mass from both sides of the equation
velocity^2 / radius = gravitational acceleration
Step 6: Solve for the velocity
velocity^2 = radius * gravitational acceleration
Step 7: Take the square root to find the velocity
velocity = √(radius * gravitational acceleration)
The acceleration due to gravity is approximately 9.8 m/s².
Step 8: Calculate the velocity
velocity = √(7.0 m * 9.8 m/s²)
velocity ≈ √68.6 m²/s²
velocity ≈ 8.28 m/s
Therefore, the child is running at a speed of approximately 8.28 m/s when they are about to lose contact with the floor.
To find how fast the child is running when about to lose contact with the floor, we need to analyze the forces acting on the child at that moment and use the concept of centripetal force.
Let's break down the problem step by step:
1. The child is running in a circle of radius 7.0m. This means the radius of the circle is 7.0m.
2. The length of the rope is 13.0m. This length represents the circumference of the circle the child is running in. Since the circumference of a circle is given by 2πr (where r is the radius), we can write the equation as:
13.0m = 2π * 7.0m (equation 1)
3. Rearranging equation 1 to solve for π:
π = 13.0m / (2 * 7.0m) (equation 2)
4. Now, we can find the circumference of the circle and the distance the child runs in one complete revolution. The distance the child runs is equal to the circumference, which is 2πr (where r is the radius):
Distance = 2π * 7.0m
5. The time it takes for the child to run one complete revolution is equal to the distance divided by the speed at which the child is running. Let's call this speed "v":
Time = Distance / v (equation 3)
6. We are interested in finding the speed at the point when the child is about to lose contact with the floor. At this moment, the centripetal force required to keep the child moving in a circle is equal to the gravitational force pulling the child downward. Mathematically, we can write:
Centripetal Force = Gravitational Force (equation 4)
7. The centripetal force is given by the formula F = mv²/r, where "m" is the mass of the child, "v" is the speed, and "r" is the radius. Rearranging this equation, we have:
mv² = F * r (equation 5)
8. The gravitational force acting on the child is given by the formula F = mg, where "m" is the mass of the child and "g" is the acceleration due to gravity. Substituting this into equation 5, we get:
mv² = mg * r (equation 6)
9. Rearranging equation 6 to solve for the speed "v":
v = √(g * r) (equation 7)
10. Now we can substitute the known values from the problem into equation 7:
v = √(9.8m/s² * 7.0m)
11. Calculate the value of "v" using a calculator:
v ≈ 9.8m/s
Therefore, the child is running at a speed of about 9.8m/s when about to lose contact with the floor.