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2. A bead of mass m is free to slide along a frictionless wire bent in...
5. A bead of mass m is free to slide on a frictionless wire bent in...
5. A bead of mass m is free to slide on a frictionless wire bent in the shape of a cosine curve y - a cos (k), where a and b are constant. Gravity points in the negative y direction. Suppose the bead starts at rest at the top of a peak. a. Find the radius of curvature of the point at the bottom of a trough. b. Find the tangential and normal components of the acceleration of the bead...
4. A bead of mass m slides on a frictionless wire bent into the shape of...
4. A bead of mass m slides on a frictionless wire bent into the shape of a parabola 2 yd as shown above. Gravity acts in the negative y direction. A spring with elastic constant k and rest length d/2 connects the bead to a fixed anchor at the point (0, -d). Find the frequency of small oscillations about equilibrium. Hint: Find the potential energy Uof the bead. Then expand Uin series, keeping only the leading x2 term, to obtain...
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 small bead with a mass m = 15.0 g slides along the frictionless wire form...
A small bead with a mass m = 15.0 g slides along the
frictionless wire form shown in the figure. The three heights
hA = 7.70 m, hB = 5.50 m,
and hC = 2.90 m are all measured from the
floor. The bead is released from rest at point A.
a) What is the speed of the bead at points B and C?
vB
= ____ m/s
vC
= ____ m/s
(b) What is the net work done on...
A small bead of mass m is free to slide along a long, thin, frictionless rod,...
A small bead of mass m is free to slide along a long, thin, frictionless rod, which spins in a horizontal plane abut one end at a frequency of f (i.e., f revolutions per second). Show that the displacement of the bead from the center of rotation as a function of time t is given by r(t) = A exp(ct) + B exp(–Ct). Find the expression for the constant C. Also, how would you determine A and B?
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...
I need to rescale (4) from the first page to the equation on the second page. 2.[60pts.] A bead of mass m is constrained to slide along a straight rigid horizontal wire. A spring with natural len...
I
need to rescale (4) from the first page to the equation on the
second page.
2.[60pts.] A bead of mass m is constrained to slide along a straight rigid horizontal wire. A spring with natural length Lo and spring constant k is attached to the bead and to a support point a distance h from the wire. See Figure 1. Let z(t) denote the position of the bead on the wire at time t. (Note that x is measured...
A bead of mass m slides without friction along a rotating wire in the shape of...
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
1. Two boxes are stacked on a frictionless table. M. = 4kg and M, 5kg. The...
1. Two boxes are stacked on a frictionless table. M. = 4kg and M, 5kg. The coefficient of friction between the boxes is such that when a 27 N force F is applied to the lower box, the boxes start to slip relative to each other. The system is then restored to rest and force F is removed. A horizontal force F is now applied to the upper box. What is this force's maximum value in order for the two...
A small block of mass m slides along the frictionless loop the loop track shown below....
A small block of mass m slides along the frictionless loop the loop track shown below. If it starts from rest at point A, what is the speed of the block at point B? (v = squareroot (10 g R)) What is the net force acting on the block at point C? (Don't forget the gravitational force. (F = -mg (8i + j) At what height above the bottom should the block be released so that the normal force exerted...
5. A bead of mass m is free to slide on a frictionless wire bent in the shape of a cosine curve y - a cos (k), where a and b are constant. Gravity points in the negative y direction. Suppose the bead starts at rest at the top of a peak. a. Find the radius of curvature of the point at the bottom of a trough. b. Find the tangential and normal components of the acceleration of the bead...
4. A bead of mass m slides on a frictionless wire bent into the shape of a parabola 2 yd as shown above. Gravity acts in the negative y direction. A spring with elastic constant k and rest length d/2 connects the bead to a fixed anchor at the point (0, -d). Find the frequency of small oscillations about equilibrium. Hint: Find the potential energy Uof the bead. Then expand Uin series, keeping only the leading x2 term, to obtain...
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 small bead with a mass m = 15.0 g slides along the
frictionless wire form shown in the figure. The three heights
hA = 7.70 m, hB = 5.50 m,
and hC = 2.90 m are all measured from the
floor. The bead is released from rest at point A.
a) What is the speed of the bead at points B and C?
vB
= ____ m/s
vC
= ____ m/s
(b) What is the net work done on...
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...
I
need to rescale (4) from the first page to the equation on the
second page.
2.[60pts.] A bead of mass m is constrained to slide along a straight rigid horizontal wire. A spring with natural length Lo and spring constant k is attached to the bead and to a support point a distance h from the wire. See Figure 1. Let z(t) denote the position of the bead on the wire at time t. (Note that x is measured...
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
1. Two boxes are stacked on a frictionless table. M. = 4kg and M, 5kg. The coefficient of friction between the boxes is such that when a 27 N force F is applied to the lower box, the boxes start to slip relative to each other. The system is then restored to rest and force F is removed. A horizontal force F is now applied to the upper box. What is this force's maximum value in order for the two...
A small block of mass m slides along the frictionless loop the loop track shown below. If it starts from rest at point A, what is the speed of the block at point B? (v = squareroot (10 g R)) What is the net force acting on the block at point C? (Don't forget the gravitational force. (F = -mg (8i + j) At what height above the bottom should the block be released so that the normal force exerted...
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