(II) The human leg can be compared to a physical pendulum, wit...

Question

(II) The human leg can be compared to a physical pendulum, with a “natural” swinging period at which walking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same: assume 55 cm. Let the upper rod have a mass of 7.0 kg and the lower rod a mass of 4.0 kg. Calculate the natural swinging period of the system.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A traditional grandfather clock employs a compound pendulum for timekeeping, which consists of two interconnected metal rods that each swing about a fixed upper pivot point. Assume that bolt rods are equally long each with a length of 0.75 m. The mass of the upper rod is 1.5 kg and the lower rod carries a mass of 1.0 kg, compute the period of oscillation for this particular pendulum system. So that's our end goal. Our end goals were asked to compute the period of oscillation for this particular pendulum system. And that's our final answer we're ultimately trying to solve for is what is the period of this pendulum system? Awesome. We're also given some multiple choice answers and they're all in the same units of seconds. So let's read them off to see what our final answer might be. A is 1.0 B is 1.4 C is 2.0 and D is 4.1. So first off, in order to calculate the period of oscillation for a compound pendulum consisting of two interconnected metal rods, we will first need to consider the physical properties of the pendulum system, including the length of the rods, their mass distribution and how they are connected. So, at this point, we now need to recall and use the equation to solve for the period of a physical pendulum. So the equation for the period of a physical pendulum states that t the period is equal to two multiplied by pi multiplied by the square root of I divided by M multiplied by G multiplied by H Awesome. Where I, in this case is the moment of inertia, M is the total mass which let's make a little, let's take a moment here to pause and say that M which is the total mass is equal to 1.5 kilograms plus 1.0 kg, which is equal to 2.5 kg. So that's the total mass is 2.5 kg. We also know that G is the acceleration due to gravity and H in this case is the center of mass. So now at this point, we need to solve for the individual values for the moments of inertia. So let us recall and use the moment of inertia equations for the upper and lower rods. So let's use the letter U to denote the subscript for upper and the subscript lowercase L to denote the lower rod. So the moment of inertia for the upper rod, which we're gonna denote as IU is equal to one third multiplied by mu multiplied by L square is equal to one third multiplied by 1.5 kg. So now we're plugging in our known variables into our equation to solve the numerical value multiplied by L squared where L squared is the length squared. So 0.75 m is the length of our pendulums. And this is squared which is equal to when we plug that into our calculator, 0.28125. And our units are kilograms multiplied by meters squared. So following are similar, we're gonna and also, so we're using lowercase L for our length of our rod. Let's just use a capital L. So we don't get confused. So this is the moment of inertia for the lower rod. So il is equal to 1/12 multiplied by M L multiplied by L squared plus M L multiplied by three halves multiplied by L squared, which is equal to 112, multiplied by 1.0 kg, multiplied by 0.75 meters squared plus 1.0 kilograms multiplied by three halves multiplied by 0.75 meters. And I accidentally put the squared on the L but I meant to put the squared on the outer. So the whole expression which is three halves multiplied by L to the power two. So this is all to the power of two. So when we plug that into our calculator, we will find that IL is equal to one point 3125 and its units once again are kilograms multiplied by meters squared. Awesome. So now we need to find the total moment of inertia. So it, which is equal to I U plus IL which is equal to when we plug that into our calculators, 1.59375. And it's gonna be in units of kilograms multiplied by meters square. So once again, we just took our moment of inertia for the upper and lower rods and we added them together to get 1.59375 kg multiplied by meters square. So now at this point, we need to recall and use the equation to solve for the center of mass H of the system. So let's recall that H is equal to M you multiplied by one half multiplied by L plus ML, multiplied by. So multiply it by. So we need to take this and we need to multiply it by three halves, multiplied by L and this is all divided by mu plus ML Awesome. So moving right along here, so we now need to plug in all of our known variables to solve for H. So when we do, we will get that H is equal to 1.5 kg multiplied by one half multiplied by zero 0.75 m plus 1.1 0.0 means this is our mass of our lower rod which is 1.0 kg multiplied by rehabs multiplied by L which the length of L. So we need to plug in our value for L which we know that L is equal to 0.75. So 0.75 0.75 m. Awesome. Perfect. And this is all divided by 1.5 kilograms plus 1.0 kg. Awesome. So let me plug that into our calculator. We will find that H is equal to 0.675 m when we round the three decimal places. So now we at this point, we need to solve for the period of oscillation T which is our final answer by recalling and using the period equation once again. So in this case, we need to note that P is equal to two multiplied by pi multiplied by the square root of it means it's the total moment of inertia divided by M multiplied by G multiplied by H which now we can plug in all of our known variables to solve for t our final answer. So we need to take two multiplied by pi multiplied by the square root of our total moment of inertia which was one point 59375 and its units are kilograms multiplied by meters squared. Awesome. And we need to divide this by our total mass which was 2.5 kilograms. And we need to multiply it by G which the acceleration due to gravity is 9.81 m per second squared. And we need to multiply this by H value which we determined our H value to be 0.675 m. Awesome. So when we plug that indoor calculator, we will find that T is equal to 1.9 seconds, which is approximately equal to 2.0 seconds. And that's our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer as to be the letter C 2.0 seconds. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

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