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You are designing a part for a piece of machinery. The part consists of a piece of sheet metal cut as shown below. The shape of the upper edge of the part is given by y₁(x), and the shape of the lower edge of the part is given by y₂(x). Y₁(x) = h(z) ² Y2(x) = h(z) ¹ where h = 6.6 m and d = 3.4 m Y₁(x) ta dm dx h d -Y/₂(x) You decide to find the moment of inertia of the part about that y axis first. The mass density per area for the sheet metal is 3 kg/m^2. In order to find the moment of inertia, first you must chop the part into small mass elements, dm's, that you know the moments of inertia for, dl's. Then you must use an integral to sum up all of the dl's.

You are designing a part for a piece of machinery. The part consists of a piece of sheet metal cut as shown below. The shape of the upper edge of the part is given by y₁(x), and the shape of the lower edge of the part is given by y₂(x). Y₁(x) = h(z) ² Y2(x) = h(z) ¹ where h = 6.6 m and d = 3.4 m Y₁(x) ta dm dx h d -Y/₂(x) You decide to find the moment of inertia of the part about that y axis first. The mass density per area for the sheet metal is 3 kg/m^2. In order to find the moment of inertia, first you must chop the part into small mass elements, dm's, that you know the moments of inertia for, dl's. Then you must use an integral to sum up all of the dl's.

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need help with these last four questions or at least some guideance on how to do them thanks. 

You are designing a part for a piece of machinery. The part consists of a piece of sheet metal cut as shown below. The shape of the upper edge of the part is given by y₁(x), and the shape of the lower edge of the part is given by y₂(x). Y₁(x) = h ( z ) ¹ 4 Y2(x) = h ( ² ) ¹ where h = 6.6 m and d h = 3.4 m - y₁(x) La dm dx d -Y₂(x) X You decide to find the moment of inertia of the part about that y axis first. The mass density per area for the sheet metal is 3 kg/m^2. In order to find the moment of inertia, first you must chop the part into small mass elements, dm's, that you know the moments of inertia for, dl's. Then you must use an integral to sum up all of the dl's.
Transcribed Image Text:You are designing a part for a piece of machinery. The part consists of a piece of sheet metal cut as shown below. The shape of the upper edge of the part is given by y₁(x), and the shape of the lower edge of the part is given by y₂(x). Y₁(x) = h ( z ) ¹ 4 Y2(x) = h ( ² ) ¹ where h = 6.6 m and d h = 3.4 m - y₁(x) La dm dx d -Y₂(x) X You decide to find the moment of inertia of the part about that y axis first. The mass density per area for the sheet metal is 3 kg/m^2. In order to find the moment of inertia, first you must chop the part into small mass elements, dm's, that you know the moments of inertia for, dl's. Then you must use an integral to sum up all of the dl's.
Moment of Inertia (y) Now find the total moment of inertia by adding up (integrating) all of the dl's. To do this, put everything in terms of x and compute the integral over x. Don't forget about limits. Then plug in the values for your constants so that you can get a value for the moment of inertia. Enter the numerical value for your Iy below. A Hint About the Limits of Integration A Hint About Your Integral Iy = 505.6kg*m^2 Total Mass Find the total mass of the part by adding up (integrating) all of the dm's. To do this, put everything in terms of x and compute the integral over x. Don't forget about limits. Then plug in the values for your constants so that you can get a numerical value for the total mass. m = 18.96kg Center of Mass (x) Find the x position of the center of mass of the part. You may need to look up the definition for the center of mass of a continuous object if you do not remember it. Here are the steps that are involved in the calculation of the center of mass. Break the object into a bunch of little dm's like you did for the moment of inertia. Multiply the dm by the x coordinate of its center of mass. Let's call this cm,dm. Add up all of the dm, integrate x cm,dm. Divide by the total mass. x cm Moment of Inertia (x) 0.457m Now you will find the total moment of inertia about the x axis. There are at least two ways to find this. You can try and find it by using an integral with respect to x. This way turns out to by much more difficult than setting up an integral expression with respect to y instead. If you set up this integral with respect to y, you should notice that the expression for the moment of inertia about the x axis is very similar to the expression for the moment of inertia about the y axis. If you think about this in the correct way, then you can actually figure this integral out with the work you have already done for Iy and not have to integrate again. Remember that the integration variable is arbitrary for a definite integral, as it is summed over when calculating the integral. So you are allowed to make the replacement y →x. Also, if you make the replacements h →d and d→ h, then the expression for I would be exactly the same as that for Iy. So you should be able to make these replacements in your final equation for Iy to give you the final equation for I. Enter the numerical value for your I below. Iz =
Transcribed Image Text:Moment of Inertia (y) Now find the total moment of inertia by adding up (integrating) all of the dl's. To do this, put everything in terms of x and compute the integral over x. Don't forget about limits. Then plug in the values for your constants so that you can get a value for the moment of inertia. Enter the numerical value for your Iy below. A Hint About the Limits of Integration A Hint About Your Integral Iy = 505.6kg*m^2 Total Mass Find the total mass of the part by adding up (integrating) all of the dm's. To do this, put everything in terms of x and compute the integral over x. Don't forget about limits. Then plug in the values for your constants so that you can get a numerical value for the total mass. m = 18.96kg Center of Mass (x) Find the x position of the center of mass of the part. You may need to look up the definition for the center of mass of a continuous object if you do not remember it. Here are the steps that are involved in the calculation of the center of mass. Break the object into a bunch of little dm's like you did for the moment of inertia. Multiply the dm by the x coordinate of its center of mass. Let's call this cm,dm. Add up all of the dm, integrate x cm,dm. Divide by the total mass. x cm Moment of Inertia (x) 0.457m Now you will find the total moment of inertia about the x axis. There are at least two ways to find this. You can try and find it by using an integral with respect to x. This way turns out to by much more difficult than setting up an integral expression with respect to y instead. If you set up this integral with respect to y, you should notice that the expression for the moment of inertia about the x axis is very similar to the expression for the moment of inertia about the y axis. If you think about this in the correct way, then you can actually figure this integral out with the work you have already done for Iy and not have to integrate again. Remember that the integration variable is arbitrary for a definite integral, as it is summed over when calculating the integral. So you are allowed to make the replacement y →x. Also, if you make the replacements h →d and d→ h, then the expression for I would be exactly the same as that for Iy. So you should be able to make these replacements in your final equation for Iy to give you the final equation for I. Enter the numerical value for your I below. Iz =
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