Mass and Angular Acceleration

A small cube (m=0.690 kg) is at a height of 297.0 cm up a frictionless track which has a loop of radius, R = 38.61 cm at the bottom. The cube starts from rest and slides freely down the ramp and around the loop. Find the velocity of the block when it is at the top of the loop.

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b) A uniform solid cylinder (m=0.690 kg, of small radius) is at the top of a similar ramp, which has friction. The cylinder starts from rest and rolls down the ramp without sliding and goes around the loop. Find the velocity of the cylinder at the top of the loop.

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Problem 3

A beam is supported only at one point, called the pivot point, as shown in the diagram. A block with mass m1 sits at the left end of the beam, a distance L1 from the pivot point. A block with mass m2 sits at the right end of the beam, a distance L2 from the pivot point. L2 > L1. Calculate all torques about the pivot point, remembering that positive is anti-clockwise. Select Yes, No, Less than, Equal to, or Cannot tell.

If m1 * L2 = m2 * L1, is there a negative torque? Given particular values of L1, L2, and m1, is it always possible to choose m2 such that the masses have no angular acceleration? For m1 = m2, does the angular acceleration depend only on L1 / L2 ? (If it depends on the actual values of L1 and L2, put ‘no’.) If m1 = m2, will the masses have an angular acceleration?

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If L1 = 0.490 m, L2 = 1.18m, m1 = 4.10 kg, and m2 = 3.40 kg, what is the angular acceleration of the beam?
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Problem 4

A 1.42 kg particle moves in the xy plane with a velocity of v = (4.26i – 3.43j) m/s. Determine the particle’s angular momentum when its position vector is r = (1.49i + 2.36j) m. Enter the k-component of the angular momentum with correct units.

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Problem 5

On a frictionless table, a glob of clay of mass 0.760 kg strikes a bar of mass 0.740 kg perpendicularly at a point 0.290 m from the center of the bar and sticks to it.

a) If the bar is 1.460 m long and the clay is moving at 5.400 m/s before striking the bar, what is the final speed of the center of mass?
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b) At what angular speed does the bar/clay system rotate about its center of mass after the impact (in rad/s)?
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Problem 6

The mass of a star is 1.67×1031 kg and it performs one rotation in 29.1 days. Find its new period (in days) if the diameter suddenly shrinks to 0.930 times its present size. Assume a uniform mass distribution before and after.

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Problem 7

A large horizontal circular platform (M = 122.1 kg, r = 3.39 m) rotates about a frictionless vertical axle. A student (m=89.3 kg) walks slowly from the rim of the platform toward the center. The angular velocity ω of the system is 3.30 rad/s when the student is at the rim. Find ω (in rad/s) when the student is 2.65 m from the center.

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Problem 8

Select True or False for each statement.

Work can be done in the absence of motion.
More power is required while slowly lifting a box than while lifting it up quickly. Work is done when the form of energy changes.
Without friction, and in the absence of external forces, the sum of the potential and kinetic energies of a body is constant. Energy conservation law for a projectile (no friction): Potential energy decrease equals the kinetic energy increase. Energy is required to do work.

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Problem 9

A large plate is balanced at its center and two students of equal mass stand at its center. The plate is rotated on a frictionless pivot about an axis through its center and perpendicular to its face. The students then begin to walk out towards opposite edges. (Select True, False, Increases, Decreases, or Stays the same). The students produce no net torque on the plate. If the students walk at a constant rate, then the plate will have a constant angular acceleration. The total angular momentum of the system … as the students walk outward. The rate of rotation … as the students walk outward.