A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.88 m. With the string hanging vertically, the object is given an initial velocity of 1.7 m/s parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.
To find the angle θ, we can consider the conservation of mechanical energy in the system. At the highest point of the pendulum's arc, the object momentarily comes to rest. At this point, all its initial kinetic energy is converted into gravitational potential energy.
1. First, let's find the initial kinetic energy (KE).
The kinetic energy is given by the formula:
KE = 0.5 * mass * velocity^2
In this case, since the mass is not given, we can temporarily assume it to be 1 kg to simplify the calculations. Therefore:
KE = 0.5 * 1 kg * (1.7 m/s)^2
= 1.445 J (Joules)
2. Next, let's find the gravitational potential energy (PE) at the highest point.
The potential energy is given by the formula:
PE = mass * g * height
Assuming the height at the highest point is h and the acceleration due to gravity is g = 9.8 m/s^2, then the potential energy becomes:
PE = 1 kg * 9.8 m/s^2 * h
However, we need to express h in terms of θ. It can be expressed as:
h = l - l * cos(θ)
Where l is the length of the string (0.88 m) and θ is the angle made by the string with its initial vertical orientation.
By substituting the equation above into the potential energy equation:
PE = 1 kg * 9.8 m/s^2 * (l - l * cos(θ))
3. Since the kinetic energy at the highest point is completely converted into potential energy, we can set KE equal to PE and solve for θ:
1.445 J = 1 kg * 9.8 m/s^2 * (l - l * cos(θ))
Simplifying the equation:
1.445 = 9.8 * (0.88 - 0.88 * cos(θ))
1.445/9.8 = 0.88 - 0.88 * cos(θ)
Rearranging the equation to isolate cos(θ):
cos(θ) = 1 - (1.445/9.8) / 0.88
Taking the inverse cosine (arccos) on both sides to find θ:
θ ≈ arccos(1 - (1.445/9.8) / 0.88)
Using a calculator, we can evaluate:
θ ≈ 64.53 degrees
Therefore, the angle θ is approximately 64.53 degrees.
To find the angle θ, we can use conservation of mechanical energy.
The mechanical energy of the pendulum is conserved if we ignore any friction or air resistance. At the highest point, the kinetic energy is zero, and all the energy is in the form of potential energy. At the lowest point, all the energy is in the form of kinetic energy.
The potential energy of the pendulum at the highest point is given by:
PE = mgh
where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above its lowest point.
The kinetic energy of the pendulum at the lowest point is given by:
KE = (1/2)mv^2
where m is the mass of the object and v is the velocity of the object at the lowest point.
Since the mechanical energy is conserved, we can equate the potential energy at the highest point to the kinetic energy at the lowest point:
mgh = (1/2)mv^2
Canceling out the mass, we get:
gh = (1/2)v^2
We can solve this equation for h:
h = (1/2) * (v^2/g)
Substituting the given values:
h = (1/2) * (1.7^2/9.8)
h = 0.145 m
Now, we can find the length of the horizontal component of the string using the equation:
l = L - h
where L is the length of the string.
Substituting the given values:
l = 0.88 - 0.145
l = 0.735 m
Finally, we can find the angle θ using trigonometry. The sine of θ is given by:
sin(θ) = (l/h)
θ = sin^(-1)(l/h)
θ = sin^(-1)(0.735/0.145)
θ ≈ 78.5 degrees
Therefore, the angle θ is approximately 78.5 degrees.