1. The problem statement, all variables and given/known data Here's the question: Imagine that you are a gymnast on a high bar, 180 degrees from the right horizontal in a fully extended position (NO angular velocity at this point). Assuming that the center of mass of your body is 1.288m from your extended hands, and that your radius of gyration is 1.4m from your extended hands.. Calculate the torque created about the bar beginning at the start and for every 30 degrees until the rotation stops or changes direction. Assume that the friction from the bar in your hands produces a constant torque value of 30Nm. My mass = 91.6kg or 898.6N 3. The attempt at a solution Okay, so.. I know that T = Fxd, but I don't know which force or distance to use. Do I use F=898.6N, which is my weight? And my center of mass which is 1.288m? Also, I'm not really sure how to solve for the different angles that I (the gymnast) am rotating around the bar at. I found a similar question to mine on a different forum, and here's how this person solved it: (I plugged in my values instead of his) sum of torque at the start (0 degrees)= 0 0=(898.6*1.288)+(-30Nm) =1127.392Nm ------------------------------------------------ sum of torque at 30 degrees from horizontal = 0 0=(898.6*(1.288cos30))+(-30Nm) =972.33Nm ------------------------------------------------ sum of torque at 60 degrees from horizontal = 0 0=(898.6*(1.288cos60))+(-30Nm) =548.7Nm ------------------------------------------------ sum of torque at 90 degrees from horizontal = 0 0=(898.6*(1.288cos90))+(-30Nm) =-30Nm ..so this is the point where the gymnast changes direction and starts ascending back up the other way. I just don't understand why this person used cos in his equations and how I am supposed to break up the different components of this problem to then create a triangle that I can solve from.. so I have all of my answers, thanks to the post from this other forum, but I have no idea how he got to this point. Any help at all would be appreciated!
Yes, the force of gravity effectively works through the center of gravity of the body. Rather nasty bit of math or poor notation there. A sum can't both be zero and have a nonzero value. Also, since there is obviously acceleration happening, the net torque cannot be zero. Draw a diagram of a case where the angle is somewhere between horizontal and vertical. What direction does the force of gravity have? What angle does this force make with the radius vector from the bar to the point of application? How would you calculate the torque it produces about the bar?
okay, I completely agree with you that my notation was wrong.. so I drew out a diagram and I now understand where the use of cosine came from and what values I was looking for, but there is a part 2 to this question.. ---------------------------------------------------------------------------- Calculate the angular velocity at each position assuming that the torque from the previous position was applied for a period of 0.1 seconds. so my torque values are as follows: at 0 degrees from horizontal, T = 1127.392Nm at 30 degrees from horizontal, T= 972.33Nm at 60 degrees from horizontal, T = 548.696Nm and at 90 degrees from horizontal, T= -30Nm so to find w (angular velocity), I attempted to first find a (angular acceleration), since a=Δw/Δt I found a from T(torque)/I(moment of inertia) My moment of inertia (I) is 176.46 --- so a at 0°=1127.392Nm/176.46 = 6.389 I don't know what the units are here! deg/s^2? rad/s^2? m/s^2 a at 30°=972.33Nm/176.46 = 5.51 a at 60°=548.696Nm/176.46 = 3.11 a at 90°=(-30Nm)/176.46 = -0.17 and from these accelerations (if I even did this part correctly), I figured I could plug them into the equation a=Δw/Δt so w=aΔt (is it okay to disregard the Δ "change in" here? or no?) w at 0°=6.389*0.1 = 0.6389 **units?** w at 30°=5.51*0.1 = 0.551 is it okay to use t=0.1s or do I use t=0.2s? w at 60°=3.11*0.1 = 0.311 w at 90°=-0.17*0.1 = -0.017 I see in my notes that the units are supposed to be in rad/s, but I'm not sure what units I even have right here, so I can't even begin to convert these numbers if I needed to.. please let me know if I did anything right/wrong.. I am feeling very lost here.. thank you!
The normal units of angular acceleration are rad/sec^{2}. A torque in N m divided by a moment of inertia in kg m^{2} will give an acceleration [itex]\alpha[/itex] in radians/sec^{2}. Degrees/sec^{2} are acceptable if the problem is using degrees for its angle measurements, but you need to be a bit careful about when they 'pop out' naturally in an equation and when you'll need to apply a conversion. You should be sure to make clear what units you use. I think your moment of inertia should be a tad higher than the 176.46 kg m^{2} you've stated. I see it being closer to 180 kg m^{2}. The equations of motion in the angular domain are a direct parallel to those in the linear domain. So if you have a moment of inertia (mass) and a force (torque), then you you can determine the angular acceleration (acceleration). a = f/m ---> [itex]\alpha = \tau/I[/itex] . Similarly, given an acceleration and a time interval, in the linear motion domain you would write [itex]v = v_i + a \Delta t[/itex], while in the angular domain you would write [itex] \omega = \omega_i + \alpha \Delta t [/itex]. Part 2 of the problem, as you describe it, is a bit odd. Still, is is what it is I would point out that for the initial position (when the body is initially horizontal) that there is no 'previous position' from which a torque, and thus an acceleration, can be obtained. But for the rest of the positions you can certainly calculate the 'legacy' torques and angular velocities. Given an initial angular velocity and torque, then [itex] \omega = \omega_o + \alpha t[/itex]. So this is the equation you should be using to find the 'current' angular velocity after the last position, given that the specified torque is applied for 0.1 seconds. The angular velocities should be cumulative. In other words, you need to apply each velocity change to the existing velocity as you go.
so you are saying that at 0° from horizontal, I should have an angular acceleration and an angular velocity of 0? ..and then for 30° from horizontal, when I use the equation ω=ω_{o} + αt, the ω_{o} would = 0? .. and sorry, the time I use would always be 0.1s, or do I need to add 0.1s every time I calculate the angular velocity? my moment of inertia might be a bit off because he told use to use a radius of gyration that is 10cm further from our hands than our COM.. so I used the formula I=mk^2 to find this. When I did all of my cos calculations, my calculator was in degrees, so I'm assuming that my accelerations are in deg/s^2.. and so I should convert all of my accelerations/velocities to rad/s^2 and rad/s..
Yes. Yes, at 30° the ω_{o} would be 0. And the time (which is really a Δt) is 0.1 every time. Given r_{g} = 1.4m, and m = 91.6kg, then I = m r_{g}^{2} = 179.5 kg m^{2}. The cosine of an angle is the same regardless of whether you use degrees or radians (if your calculator is set for the corresponding units!). It's the same angle after all, and a cosine is really nothing more than a ratio of triangle side lengths which is independent of the units used to express the angle. When you calculate an acceleration by dividing the torque by the moment of inertia, the result will be in radians. Radians are the "natural" units for angles that result from such calculations. You'd have to convert these results to degrees if you want degrees. It's generally easier to just leave everything in radians until a final result is to be presented using degrees as the angular unit.
okay, thank you so much for your help.. and I just realized, I forgot that I rounded my radius of gyration in this forum to 1.4m.. it is really 1.388 in my notes, and that calculation works out. also, since α=T/I, I am finding an angular acceleration of 6.388rad/s^2 at the 0° from horizontal point.. how is it possible to have acceleration when there is no velocity? so when I calculate the angular velocity for each level, for example if I am calculating it at 30° from horizontal, would I use the α from 30° or the α from 0°? (angular accel. at 0° = 6.388rad/s^2) (angular accel. at 30° = 5.51rad/s^2) ω at 30° = 0 + (6.388rad/s^2 * 0.1s) or ω at 30° = 0 + (5.51rad/s^2 * 0.1s) one more thing: the question says to calculate torque every 30° until the rotation "stops or changes direction" and I know that a counter-clockwise rotation is considered positive, and a clockwise rotation is considered negative.. so at 90°, when the torque = -30Nm, can I say that this is the point where rotation changes direction? ...though, surely the gymnast doesn't actually start rotating backwards before she reaches the bottom....
Velocity and acceleration are two different things. You needn't have one to have the other. For example, if you throw an object straight upwards its velocity slows until it reaches the top of its trajectory, then accelerates downward as it falls back to the ground. At the top of its trajectory its velocity was zero, yet gravity was accelerating it constantly the whole time. From zero. You are told to assume that the acceleration calculated at the last position applied over 0.1 seconds in order to bring you to the new position. Just like in real life the gymnast is going to swing through the bottom position and partway up the other side of the arc. How close he comes to the horizontal on the other side will depend upon how much energy he lost on the way due to the 30Nm frictional torque. Pay attention to the signs of the forces and torques! He should be decelerating as he rises.
okay I got all of the torques and velocities for each different angle, and it turned out that I had not calculated enough angles initially.. I ended up going all the way up to 240° where the velocity became negative, so I knew that a change in direction had occurred. there is one more part to this question, and it goes like this: "recalculate the ascending phase of the swing but assume you take a pike position (flexion at the hips, bring the legs up) instantaneously at 90° (vertical), and assume that your center of mass is 20cm closer to the hands. Also, assume that your radius of gyration is 10cm further from the hands than the center of mass." so, without doing calculations here, I'm assuming that my angular velocity should increase, and my moment of inertia will decrease. The pike will decrease the moment arm, and the torque values will also decrease.. when I used my torque values to find angular acceleration, then used those values to calculate the angular velocity, my velocities were actually decreased compared to when i originally didn't pike in the ascending phase. I don't know what I'm doing wrong here...
There should be a sudden increase in his velocity when he "instantaneously" assumes the pike position. Use conservation of angular momentum to determine how the velocity changes at the 90° position.
I have a question about the first half of the question, what are you talking about with a triangle i am completely lost with that. I have the torques for the one side but do i have to do the torques for when it goes it the other side cause it states when it changes direction or stops but it is still going in that direction so would i have to find the torque also?
This doesn't seem quite right, as the angular inertia should be similar to that of a solid rod of length L rotating about one end, which would m L^2 / 3. Assuming L is about 1.288 m x 2, then angular inertia would be a bit more than 2.2 m, but using the center of mass method, the angular inertia is m r^2 = 1.66 m. Note the angular position when the pike occurs makes a big difference, the greatest change occurs if the pike is done if the gymnast is directly below the bar at the bottom of the swing, while the smallest change (or a negative change) occurs if the pike is done if the gymnast is directly above the bar at the top of the swing. When at the bottom of the swing, the angular velocity and the force (tension within the gymnast's body) is greater, so the work done by piking at the bottom of the swing is much greater than at the top (where the angular velocity could be near zero, and "negative" work done by allowing gravity to pull the legs into a pike position).