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4. A bead of mass m slides on a frictionless wire bent into the shape of...
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...
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...
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 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 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...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The circle stands up in a vertical plane and rotates about the z-axis with constant angular velocity . Write down the Lagrangian. Find the equations of motion. For an angular velocity greater than some critical angular velocity , the bead will experience small oscillations about some stable equilibrium point . Find and (). We were unable to transcribe this imageWe were unable to transcribe this...
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...
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...
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...
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 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 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...
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...
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