A child standing on a platform holds a rope taut at an angle θ to the vertical. The child swings from the platform starting from rest. When the child swings through the lowest position, the tension in the rope is 1.9 times the child's weight. Determine the value of θ.
To determine the value of θ, we need to use the concept of forces and equations of motion. Here's how we can solve the problem step by step:
Step 1: Identify the forces acting on the child at the lowest position.
At the lowest position, two forces act on the child: the tension in the rope (T) and the child's weight (W). The tension in the rope is acting towards the center of the circular motion, while the weight is acting vertically downwards.
Step 2: Write down the equations of motion for the vertical and horizontal directions.
Since the child is swinging from rest, we can assume there is no horizontal force at the lowest point (when the child is at the bottom of the swing). So, we only need to consider the vertical forces.
In the vertical direction, the equation of motion is:
ΣFy = ma
Where ΣFy is the sum of the forces in the vertical direction, m is the mass of the child, and a is the acceleration.
Step 3: Determine the value of the child's weight (W).
The weight of an object is given by the equation:
W = mg
Where m is the mass of the child and g is the acceleration due to gravity. Since the problem doesn't give the mass of the child, we can cancel it out by dividing both sides of the equation by m:
W/m = g
Step 4: Express the tension in terms of the child's weight.
The problem statement states that the tension in the rope is 1.9 times the child's weight. Therefore, we can write:
T = 1.9W
Step 5: Substitute the equations for weight and tension into the equation of motion.
Using the equations for weight and tension, we can substitute them into the equation of motion:
ΣFy = ma
T - W = ma
Substituting the values for tension (T) and weight (W), we get:
1.9W - W = ma
Step 6: Simplify the equation and solve for the acceleration (a).
Rearrange the equation to isolate the acceleration:
0.9W = ma
Step 7: Simplify the equation and solve for the angle (θ).
At the lowest position, the acceleration (a) is given by a = gsin(θ), where θ is the angle the rope makes with the vertical.
Substituting this into the previous equation, we have:
0.9W = mgsin(θ)
Now substitute the equation for weight (W = mg) into the equation:
0.9(mg) = mgsin(θ)
The mass (m) cancels out:
0.9g = gsin(θ)
Finally, divide both sides of the equation by g:
0.9 = sin(θ)
Step 8: Determine the value of θ.
To find the value of θ, we need to take the inverse sine (or arcsin) of both sides of the equation:
θ = arcsin(0.9)
Using a calculator, we find:
θ ≈ 64.26 degrees (rounded to two decimal places)
So, the value of θ is approximately 64.26 degrees.