A conical pendulum consists of a plumb bob of mass m moving in a circular path in a horizontal plane, as shown in the figure to the right. During the movement, the support wire, of length l, maintains the constant angle θ with the vertical. Show that the magnitude of the angular momentum of the plumb bob about the center of the dotted circle is L = ((m ^ 2 gl ^ 3 sin ^ 4 θ) / cosθ) ^ (1⁄3).
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A conical pendulum consists of a plumb bob of mass m moving in a circular path in a horizontal plane, as shown in the figure to the right. During the movement, the support wire, of length l, maintains the constant angle θ with the vertical. Show that the magnitude of the
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- In the figure here, a 26 kg child stands on the edge of a stationary merry-go-round of radius 1.8 m. The rotational inertia of the merry-go-round about its rotation axis is 160 kg·m2. The child catches a ball of mass 1.5 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity of magnitude 12 m/s, at angle φ = 59 ˚ with a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught?A particle of mass m in the figure below slides down a frictionless surface through height h and collides with a uniform vertical rod (of mass M and length d), sticking to it. The rod pivots about point O through an angle θ when it momentarily stops. Find θ in terms of m, M, g, h, d and various constants. (Please type answer no write by hend)A block of mass m is moving with speed v along a horizontal surface when it collides with a uniform rod of mass 2m and length L attached at one end to a pivot. The surface and pivot have negligible friction. The rod is vertical when the block collides with the end of the rod. The block sticks to the rod, and the block-rod system rotates so that the end of the rod reaches a height h, as shown above. The total rotational inertia of the rod about the pivot is 2mL/3. Express answers in parts (a), (b), and (c) in terms of m, L, v, and physical constants as appropriate. c. Derive an expression for the mechanical energy dissipated during the collision.
- A disc of radius R = 0.41 m and of mass M = 1.22 kg is rotating with the initial angular momentum ωi = 0.51 rev/s on a frictionless horizontal plane. You drop a small clay piece of mass m = 0.19 kg vertically, it lands, and is stuck on a point of the rim of the disc at the distance R = 0.41 m from the center of the disc (Fig.9). The initial moment of inertia of the disk about the axis of rotation through its center is Ii = MR2/2. What is the angular velocity of the disc-clay system in rev/s after the collision? Keep three significant figures for the answer. Note the clay piece has negligible size so that it can be treated as a point.A uniform rod of mass 210 g and length 100 cm is free to rotate in a horizontal plane around a fixed vertical axis through its center, perpendicular to its length. Two small beads, each of mass 22 g, are mounted in grooves along the rod. Initially, the two beads are held by catches on opposite sides of the rad's center, 12 cm from the axis of rotation. With the beads in this position, the rod is rotating with an angular velocity of 16.0 rad/s. When the catches are released, the beads slide outward along the rod. (a) What is the rod's angular velocity (in rad/s) when the beads reach the ends of the rod? (Indicate the direction with the sign of your answer.) rad/s (b) What is the rod's angular velocity (in rad/s) if the beads fly off the rod in a tangential, straight line path? (Indicate the direction with the sign of your answer.) rad/sAt the instant of the figure, a 7.70 kg particle P has a position vector of magnitude 6.70 m and angle 0₁ = 49.0° and a velocity vector of magnitude 8.90 m/s and angle 0₂ = 29.0°. Force F, of magnitude 9.80 N and angle 03 = 29.0° acts on P. All three vectors lie in the xy plane. About the origin, what are the magnitude of (a) the angular momentum of the particle and (b) the torque acting on the particle? (a) Number i (b) Number Units Units V. 0 0₂
- At the instant of the figure, a 1.40 kg particle P has a position vector of magnitude 2.90 m and angle 0₁ = 48.0° and a velocity vector of magnitude 7.50 m/s and angle 0₂ = 32.0°. Force F, of magnitude 6.20 N and angle 03 = 32.0° acts on P. All three vectors lie in the xy plane. About the origin, what are the magnitude of (a) the angular momentum of the particle and (b) the torque acting on the particle? (a) Number i (b) Number Units Units F03 xA planet is at a position of r→=3.0×10^12mx̂ − 4.0×10^12mŷ as it moves in an elliptical orbit around a large star at the origin (x=0, y=0). If the planet has a mass of 6.0 ×1026 kg and is moving with a velocity of v→=9.5km/sx̂, what is its angular momentum vector? (give magnitude and direction)Two thin rods of length L are rotating with the same angular speed ω (in rad/s) about axes that pass perpendicularly through one end. Rod A is massless but has a particle of mass 0.78 kg attached to its free end. Rod B has a mass of 0.78 kg, which is distributed uniformly along its length. The length of each rod is 0.71 m, and the angular speed is 4.8 rad/s. Find the kinetic energies of rod A with its attached particle and of rod B.