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2) Starting from the origin, a heavy particle slides down a frictionless wire in the x,...
A bead of mass m slides along a frictionless wire under the influence of gravity. The...
A bead of mass m slides along a frictionless wire under the influence of gravity. The shape of the wire is given by the equation y = axa, where x denotes the horizontal co-ordinate, y denotes the vertical co-ordinate, and a is a constant. (a) Use Lagrange's equation to determine the equation of motion. (b) Compute Hamilton's equations of motion and show that they are equivalent to your result for item (a).
A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from...
A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from rest at the top . The centripetal acceleration of the mass ii. The tangential acceleration of the mass degrees b. Determine the value of 8 at which the particle leaves the sphere.
A particle of mass m slides without slipping down and inclined plane of mass M. The...
A particle of mass m slides without slipping down and inclined plane of mass M. The plane(a triangle) in turn can slide along the horizontal surface without friction. Find the Lagrangian in terms of (x, s, dx/dt, ds/dt), where s is the distance the particle has traveled down the incline, and solve the corresponding Euler-Lagrangian equations. Let x(t) be the location of the block, relative to the fixed origin, and x(0) = dx/dt(0) = 0, as well as s(0) =...
A frictionless wire is bent into the shape of a cycloid curve, with coordinates given by...
A frictionless wire is bent into the shape of a cycloid curve,
with coordinates given by the parametric equations
? = ?(? + sin ?), ? = ?(1 − cos ?), for −? < ? < ?. The x
axis is horizontal, and y is vertically upwards. A bead of
mass m slides freely on the wire. Show that the distance s,
measured along the wire from the origin, is given by
? = 4? sin. Write out the potential...
5. A particle of mass m slides without friction and starting from rest down a block...
5. A particle of mass m slides without friction and starting from rest down a block of mass M in which the surface is shaped into a quadrant of a circle of radius R. The block also slides without friction on a table top. Choose two suitable generalized coordinates a) Using the Lagrange-multiplier formalism, find the equations of motion of the mass and the block and the reaction force of the block on the mass (ie. the constraint force b)...
Problem 3 10 marks A particle of mass m slides down a frictionless incline of angle...
Problem 3 10 marks A particle of mass m slides down a frictionless incline of angle a, mass M and length L which is on a horizontal frictionless plane (see the Figure). If the particle starts initially from rest at the top of the incline, prove that the time for the particle to reach the bottom is given by, 2L(M + m sina) (M + m)g sina To setup the problem, choose a fixed vertical xy coordinate system as in...
4.0 m Question 4: A 2.0 kg particle slides down from rest from position A. When...
4.0 m Question 4: A 2.0 kg particle slides down from rest from position A. When it reaches B, it enters a rough horizontal segment with f4k-0.20, and eventually comes to rest at D. The surface from A to B is frictionless, and point A is located 4.0 m above the ground (a) Draw a free body diagram (FBD) for the particle at A (b) Draw a free body diagram (FBD) for the particle at C (c) Is the total...
2. Starting from rest, a block of mass m slides down a frictionless incline at angle...
2. Starting from rest, a block of mass m slides down a frictionless incline at angle θ(0◦ < θ < 90◦) where it runs into a spring of spring constant k. When the block momentarily stops, it has compressed the spring by distance x. Find expressions for (a) the distance the block slides down the incline from when it is released to when it momentarily stops (b) the distance between the point of the first block-spring contact and the point...
2. A bead of mass m is free to slide along a frictionless wire bent in...
2. A bead of mass m is free to slide along a frictionless wire bent in the curve yx3 where a is a positive constant. The bead starts from rest at x - a and slides under the influence of a constant gravitational field g pointing in the negative y direction. Find the time required for the bead to reach the origin. Express your answer in terms of the constants a and g Hint: Use the energy method. You may...
A bead of mass m slides smoothly from point A to point B on a semicircular horizontal wire loop of radius R. It is attracted toward its starting point A by a force F directly proportional to its dist...
A bead of mass m slides smoothly from point A to point B on a
semicircular horizontal wire loop of radius R. It is attracted
toward its starting point A by a force F directly proportional to
its distance r from A (for instance, you can imagine an elastic
string connecting m to A). What it reaches B the force toward A is
Fo. Numerical values: Fo = 2 N,
R = 20 cm, m = 500 g.
Calculate the...
A bead of mass m slides along a frictionless wire under the influence of gravity. The shape of the wire is given by the equation y = axa, where x denotes the horizontal co-ordinate, y denotes the vertical co-ordinate, and a is a constant. (a) Use Lagrange's equation to determine the equation of motion. (b) Compute Hamilton's equations of motion and show that they are equivalent to your result for item (a).
A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from rest at the top . The centripetal acceleration of the mass ii. The tangential acceleration of the mass degrees b. Determine the value of 8 at which the particle leaves the sphere.
A frictionless wire is bent into the shape of a cycloid curve,
with coordinates given by the parametric equations
? = ?(? + sin ?), ? = ?(1 − cos ?), for −? < ? < ?. The x
axis is horizontal, and y is vertically upwards. A bead of
mass m slides freely on the wire. Show that the distance s,
measured along the wire from the origin, is given by
? = 4? sin. Write out the potential...
5. A particle of mass m slides without friction and starting from rest down a block of mass M in which the surface is shaped into a quadrant of a circle of radius R. The block also slides without friction on a table top. Choose two suitable generalized coordinates a) Using the Lagrange-multiplier formalism, find the equations of motion of the mass and the block and the reaction force of the block on the mass (ie. the constraint force b)...
Problem 3 10 marks A particle of mass m slides down a frictionless incline of angle a, mass M and length L which is on a horizontal frictionless plane (see the Figure). If the particle starts initially from rest at the top of the incline, prove that the time for the particle to reach the bottom is given by, 2L(M + m sina) (M + m)g sina To setup the problem, choose a fixed vertical xy coordinate system as in...
4.0 m Question 4: A 2.0 kg particle slides down from rest from position A. When it reaches B, it enters a rough horizontal segment with f4k-0.20, and eventually comes to rest at D. The surface from A to B is frictionless, and point A is located 4.0 m above the ground (a) Draw a free body diagram (FBD) for the particle at A (b) Draw a free body diagram (FBD) for the particle at C (c) Is the total...
2. A bead of mass m is free to slide along a frictionless wire bent in the curve yx3 where a is a positive constant. The bead starts from rest at x - a and slides under the influence of a constant gravitational field g pointing in the negative y direction. Find the time required for the bead to reach the origin. Express your answer in terms of the constants a and g Hint: Use the energy method. You may...
A bead of mass m slides smoothly from point A to point B on a
semicircular horizontal wire loop of radius R. It is attracted
toward its starting point A by a force F directly proportional to
its distance r from A (for instance, you can imagine an elastic
string connecting m to A). What it reaches B the force toward A is
Fo. Numerical values: Fo = 2 N,
R = 20 cm, m = 500 g.
Calculate the...
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