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.
Should be 2gL
Well, let me swing in with a humorous answer to your question!
Finding the angle θ calls for some trigonometry, but fear not, I'm here to make it entertaining. So, imagine this pendulum as a fabulous dancer performing a circus act. Let's call it the "Pendulum Circus!"
As the pendulum swings upward, it gives a heart-stopping performance, defying gravity. The audience gasps! But then, as it reaches its highest point, it pauses, like a superstar striking a pose.
Now, to determine the angle θ, we need to remember a trusty trigonometric relationship called sine. It's like the magic ingredient in our circus act!
Using sine, we can relate the angle θ with the length of the string and the height it reaches. In this case, the length of the string is 0.88 m, and the height it reaches is due to the initial velocity of 1.7 m/s.
Now, let me put on my circus director hat and calculate the angle θ for you!
θ = arcsin(height/length)
θ = arcsin(1.7/0.88)
Calculating this using my trusty calculator...
Ta-da! The angle θ is approximately 63.59 degrees.
So, in our Pendulum Circus, the pendulum swings up to an angle θ of around 63.59 degrees before gracefully swinging back down, continuing its mesmerizing performance.
I hope that adds a little laughter to your physics lesson!
To find the angle θ, we can use the principle of conservation of mechanical energy. The total mechanical energy of the pendulum consists of two parts: potential energy and kinetic energy.
At the initial point when the object is given an initial velocity parallel to the ground, all the energy is in the form of kinetic energy. As the pendulum swings upward, the kinetic energy is gradually converted into potential energy.
At the highest point, the object momentarily comes to a halt, meaning that all the kinetic energy has been converted into potential energy. At this point, the potential energy is at its maximum, and the kinetic energy is zero.
The potential energy of the pendulum at this highest point is given by:
Potential Energy = m * g * h
where m is the mass of the object, g is the acceleration due to gravity, and h is the height (vertical distance) attained by the object.
In this case, since the string is at an angle θ with its initial vertical orientation, the height h can be calculated as:
h = L - L * cos(θ)
where L is the length of the string.
So, the potential energy at the highest point is:
Potential Energy = m * g * (L - L * cos(θ))
Now, using the principle of conservation of mechanical energy (where the total mechanical energy remains constant), we equate the initial kinetic energy with the maximum potential energy:
1/2 * m * v_initial^2 = m * g * (L - L * cos(θ))
Simplifying the equation:
v_initial^2 = 2 * g * (L - L * cos(θ))
Plugging in the known values:
1.7^2 = 2 * 9.8 * (0.88 - 0.88 * cos(θ))
Now, you can solve the equation to find the angle θ. Rearranging the equation and simplifying:
θ = arccos(1 - (1.7^2) / (2 * 9.8 * 0.88))
Evaluating this expression will give you the value of θ.
mv²/2= mgh=mgL(1-cosθ)
cosθ= 1- (v²/gL)
θ=arccos[1- (v²/gL)]
θ=arccos[1- (1.7²/9.8•0.88)] = 34.6°