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URGENT: We consider a cart of mass M that is running on wobbly wheels. To explore the consequences of the wobbling on the potential energy we assume that the wheels take the form of ellipses, and that they both have the same form. Adopting polar coordinates the form is described by q = R()r() (red arrow in the sketch to the right) with..

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We considet a nart of mass. M that in running on woblaly whoek. To exploav the coesyornose af the woldling on the potratial enedgy wn assume that the whords take the form of elligien, and that they both haw the same form. Adogiting polar coordinates the (red anow in the saitch to the right ) with R())=14cosR 3 O for than poitina, abd aperify whikh valoc of r was whopted. Hike does the shape changer utum adegeing acother value of E For which value: b) The green arroe shows the arrow n~() that is orthogual tu the whod surfice at position . Show that in talow the Joblioniog form n()=2(r()+1coscsin(y)) Determine L and c in this erpenosin 2. Shin that n~()=e()=0 c) Let be the pinition whre the wherl is lewhing the grousht. How does the petential enersy of the nuat depend on the crientation of of the wheel? Hint: You may whime that the nues M af the cart is dhminated to its axle. d) What is the smallost foece reguird to boll the caut at a given poitice ? What will be the dirnction of thas Seret What hapgens when you purh in asoeber deretion? Problem 7.4. Potential energy of a cart with a wobbly wheel We consider a cart of mass M that is running on wobbly wheels. To explore the consequences of the wobbling on the potential energy we assume that the wheels take the form of ellipses, and that they both have the same form. Adopting polar coordinates the form is described by q=R()r^() (red arrow in the sketch to the right) with R()=1cosR0 a) In the sketch the length scales are made dimensionless based on R0. The red arrow denotes a vector from the wheel axle to its surface. Add a label indicating for this position, and specify which value of was adopted. How does the shape change when adopting another value of : For which value will you obtain a circle? For which values of does one obtain ellipses? The green arrow shows the arrow n^() that is orthogonal to the wheel surface at position . Show that it takes the following form n^()=L(r^()+1coscsin^()) Determine L and c in this expression. Hint: 1. Argue that v()=dq()/d is tangential to the wheel surface. 2. Show that n^()v()=0. c) Let be the position where the wheel is touching the ground: How does the potential energy of the cart depend on the orientation of the wheel? Hint: You may assume that the mass M of the cart is dominated by its axle. d) What is the smallest force required to hold the cart at a given position ? What will be the direction of this force? What happens when you push in another direction? We considet a nart of mass. M that in running on woblaly whoek. To exploav the coesyornose af the woldling on the potratial enedgy wn assume that the whords take the form of elligien, and that they both haw the same form. Adogiting polar coordinates the (red anow in the saitch to the right ) with R())=14cosR 3 O for than poitina, abd aperify whikh valoc of r was whopted. Hike does the shape changer utum adegeing acother value of E For which value: b) The green arroe shows the arrow n~() that is orthogual tu the whod surfice at position . Show that in talow the Joblioniog form n()=2(r()+1coscsin(y)) Determine L and c in this erpenosin 2. Shin that n~()=e()=0 c) Let be the pinition whre the wherl is lewhing the grousht. How does the petential enersy of the nuat depend on the crientation of of the wheel? Hint: You may whime that the nues M af the cart is dhminated to its axle. d) What is the smallost foece reguird to boll the caut at a given poitice ? What will be the dirnction of thas Seret What hapgens when you purh in asoeber deretion? Problem 7.4. Potential energy of a cart with a wobbly wheel We consider a cart of mass M that is running on wobbly wheels. To explore the consequences of the wobbling on the potential energy we assume that the wheels take the form of ellipses, and that they both have the same form. Adopting polar coordinates the form is described by q=R()r^() (red arrow in the sketch to the right) with R()=1cosR0 a) In the sketch the length scales are made dimensionless based on R0. The red arrow denotes a vector from the wheel axle to its surface. Add a label indicating for this position, and specify which value of was adopted. How does the shape change when adopting another value of : For which value will you obtain a circle? For which values of does one obtain ellipses? The green arrow shows the arrow n^() that is orthogonal to the wheel surface at position . Show that it takes the following form n^()=L(r^()+1coscsin^()) Determine L and c in this expression. Hint: 1. Argue that v()=dq()/d is tangential to the wheel surface. 2. Show that n^()v()=0. c) Let be the position where the wheel is touching the ground: How does the potential energy of the cart depend on the orientation of the wheel? Hint: You may assume that the mass M of the cart is dominated by its axle. d) What is the smallest force required to hold the cart at a given position ? What will be the direction of this force? What happens when you push in another direction

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