A boy is whirling around a stone tied at the end of a 0.55 m long string at constant angular speed of 5 revolutions per second in a plane perpendicular to the ground. The string breaks when it is making an angle of 53 degrees above the horizontal and the stone is on the way up. The stone is 1.15 m above the ground when this happens. How long does the stone take to hit the ground after the string breaks?
i tried and I got 2.8972 seconds but it was wrong =(
Thank you!
When the stone takes off, it starts out at at angle of 37 degrees above the horizontal. (That is the complement of 53 degrees). The speed is initially
2*pi*0.55*5 = 17.28 m/s, and the vertical speed component is initially
17.28 sin 37 = 10.40 m/s
The stone spends 10.40/g = 1.06 s going up and an equal time going down, to the elevation where it started, for a total of 2.12 s in the air. You will need to add additional time to that to account for traveling from 0.55 + 0.55 sin 53 = 0.989 meters (the release altitude) to the ground. That is about 0.09 seconds more, for a total of 2.21 seconds
Thank you so much!!!
I took the sin of 53 deg instead of 37 deg
To solve this problem, we can use the kinematic equations of motion to determine the time it takes for the stone to hit the ground after the string breaks.
Let's break down the problem into two parts:
1. The time it takes for the stone to reach the maximum height after the string breaks.
2. The time it takes for the stone to fall from the maximum height to the ground.
First, let's calculate the time it takes for the stone to reach the maximum height after the string breaks:
Using the equation of motion for vertical displacement, we have:
y = y0 + v0y * t + (1/2) * a * t^2
Where:
y = final vertical position = 1.15 m
y0 = initial vertical position = 0 m (because the stone starts from the ground)
v0y = initial vertical velocity = 0 m/s (at the maximum height, the vertical velocity is 0)
a = acceleration due to gravity = -9.8 m/s^2 (negative sign because gravity acts downward)
Rearranging the equation, we get:
t = sqrt((y - y0) / (0.5 * a))
Substituting the values, we have:
t = sqrt((1.15 - 0) / (0.5 * (-9.8)))
t = sqrt(1.15 / (-4.9))
t = sqrt(-0.2347)
The value under the square root is negative, which is not physically meaningful. Therefore, the time taken to reach the maximum height is not achievable, and we need to reconsider our approach.
Instead of calculating the time to reach the maximum height, let's use the concept of conservation of mechanical energy to find a solution.
At the instant when the string breaks, the stone has a certain amount of mechanical energy due to its rotational speed:
E_initial = (1/2) * I * (w^2)
Where:
E_initial = initial mechanical energy
I = moment of inertia
w = angular speed in radians per second
The moment of inertia for a point mass rotating at the end of a string is given by:
I = m * r^2
Where:
m = mass of the stone
r = radius of the circular path = 0.55 m
Given the angular speed, we can calculate the initial mechanical energy:
E_initial = (1/2) * m * r^2 * (w^2)
E_initial = (1/2) * m * 0.55^2 * (5 * 2π)^2
Since we know the initial mechanical energy, at the maximum height, all the mechanical energy is in the form of gravitational potential energy:
E_final = m * g * h
Where:
E_final = final mechanical energy (all in the form of gravitational potential energy)
m = mass of the stone
g = acceleration due to gravity = 9.8 m/s^2 (positive because it acts upward)
h = maximum height reached by the stone = 1.15 m
Setting E_initial equal to E_final, we can solve for the mass (m):
(1/2) * m * 0.55^2 * (5 * 2π)^2 = m * 9.8 * 1.15
Canceling the mass (m) from both sides:
(1/2) * 0.55^2 * (5 * 2π)^2 = 9.8 * 1.15
Simplifying the equation:
0.30225 * (5 * 2π)^2 = 11.27
Solving for (5 * 2π)^2:
(5 * 2π)^2 = 11.27 / 0.30225
(5 * 2π)^2 = 37.286
Taking the square root of both sides:
5 * 2π = sqrt(37.286)
5 * 2π = 6.1067
Solving for 5 * 2π:
5 * 2π ≈ 6.1067
So the value of 5 * 2π is approximately equal to 6.1067.
Now, let's calculate the time it takes for the stone to fall from the maximum height to the ground:
Using the equation of motion for vertical displacement, we have:
y = y0 + v0y * t + (1/2) * a * t^2
Where:
y = final vertical position = 0 m (as the stone hits the ground)
y0 = initial vertical position = 1.15 m
v0y = initial vertical velocity = 0 m/s (at the maximum height, the vertical velocity is 0)
a = acceleration due to gravity = -9.8 m/s^2 (negative sign because gravity acts downward)
Simplifying the equation, we have:
0 = 1.15 + 0 * t + (1/2) * (-9.8) * t^2
Rearranging the equation, we get:
4.9t^2 = 1.15
t^2 = 1.15 / 4.9
t^2 = 0.2347
t = sqrt(0.2347)
Calculating the square root:
t ≈ 0.4841 seconds
Therefore, the stone takes approximately 0.4841 seconds to hit the ground after the string breaks.
Note: It's worth mentioning that you received a different answer (2.8972 seconds) because you used an incorrect approach for calculating the time taken to reach the maximum height. It's important to consider other concepts like energy conservation in this scenario.