Jeremy S. answered • 01/02/20
Physics Tutor (Experienced and Welcoming)
Hello! Pendulums (like in an old clock) can seem boring, but they also bring in many physics concepts together, including motion, force, energy, and rotation. Understanding pendulums can also be the basis for understanding more complex oscillators, such as those found in electric circuits or those studied in quantum mechanics. Let's see if we can address your question.
A) We need to make some assumptions about this pendulum. First, we are going to make an assumption about how the mass is distributed. We are going to assume that this is a simple pendulum. When your book or the teacher doesn't tell you anything specific about what the pendulum actually looks like, you should make this assumption. That means that we can approximate the pendulum as having all its mass at the far end. In other words, we are talking about a pendulum where a heavy mass is at the end of a light string. If we distribute the mass along the pendulum in some other way (Example: A stick that swings back and forth), then the period will be different. Notice that this assumption/approximation is about the distribution of the pendulum's mass, not about it's total mass. The total mass of the pendulum won't affect its period here.
The second assumption that we need to make is that the pendulum is at rest on Earth. A pendulum swings faster, and therefore has a shorter period, if the force of gravity acting on it is stronger.
Our third assumption is that this pendulum has a relatively low amplitude; it doesn't swing more than a 15 degree angle from the center point. That's realistic for most actual pendulums, and it allows us to simplify our thinking and especially our math considerably. (When an angle is less than about 15 degrees, sin(angle)≈tan(angle)≈angle in radians.)
Our fourth assumption is that this pendulum is undamped. In other words air resistance and other forces aren't doing anything to slow it down over time. Obviously, this would affect the period of the pendulum if it were present. The only significant forces acting on the pendulum bob are gravity and the string.
That might seem like a lot, but you should be very clear (especially with yourself) about the approximations that you are making when you solve a problem, and consider to what degree it is okay to make each one.
With those approximations, the period of your pendulum is given by T=2π*sqrt(L/g), where T is the period in seconds, L is the length of the pendulum, and g is the acceleration from gravity (9.8 m/s/s on Earth). This formula can be derived from considerations of Newton's 2nd law and confirmed through experiment. We won't do that here.
To answer the original question, you just plug L=1.5 m and g=9.8 m/s/s into this formula.
B) This is a question about energy and its conservation. As the pendulum is swinging from its highest point to its lowest point and back up to its highest point, energy is being transformed from gravitational potential energy (PE=mgh) to kinetic energy (KE=mv2/2) and then back to gravitational potential energy. Since the pendulum isn't colliding with anything, and we are assuming no air resistance, these are the only notable energy transformations. The wording of the problem implies that we should treat the height of the pendulum as zero at the bottom of its swing.
Then, Energybottom = Energytop → KEbottom = PEtop → mvbottom2/2 = mghtop
From there, the masses cancel, and you can solve for htop, the height of the pendulum at the top of its swing.
C) Here, you can go back to the equation T=2π*sqrt(L/g). Notice that as the strength of gravity decreases from 9.8 m/s/s to 3.59 m/s/s, that the period gets longer. You can calculate exactly how much longer by plugging in values.
As I wrote up top, there's a lot going on in this problem. It's possible to take a plug-and-chug approach to it, but you will get a lot more long term understanding if you think more deeply about what is going on.
Laveeza S.
thankyou01/03/20