Torque - Dynamical Behavior

         

The position of a 3 kg3 \text{ kg} particle moving on the xyxy-plane at time tt (in seconds) is given by the equation r=(2t2i^(4t+4t2)j^) m.\vec{r} = \left( 2 t^2 \hat{i} - (4 t + 4 t^2) \hat{j} \right) \text{ m} . Considering the origin OO as the pivot, what is the magnitude of the torque on the particle at t=6 s? t= 6 \text{ s}?

A fixed pulley-like device is used to lift a box of mass m=13 kg,m=13\text{ kg}, as shown in figure above. The outer radius of the device is R=7 m, R=7 \text{ m}, and the radius of the hub is r=2 m.r= 2 \text{ m}. When a constant horizontal force F=130 N F = 130 \text{ N} is applied to a rope wrapped around the outer rim of the device, the box hanging from a rope wrapped around the hub acquires an upward acceleration of magnitude 2 m/s2. 2 \text{ m/s}^2. Then what is the rotational inertia of the device about its axis of rotation?

The gravitational acceleration is g=10 m/s2. g= 10 \text{ m/s}^2.

A particle moves clockwise in a circle around the origin on the xy xy -plane. If its angular momentum as a function of time tt (in seconds) is 5t2 kgm2/s, 5 t^2 \text{ kg} \cdot \text{m}^2\text{/s}, what is the magnitude of the torque on the particle at t=4 s? t = 4 \text{ s}?

A 6 kg 6 \text{ kg} particle that moves on the xyxy-plane with velocity v=(6i^+2j^) m/s \vec{v} = \left( 6 \hat{i} + 2 \hat{j} \right) \text{ m/s} is at x=9 m x= 9 \text{ m} and y=12 m.y= 12 \text{ m}. It is being pulled by a 6 N 6 \text{ N} force in the x -x direction. Considering the origin as the pivot, what is the torque of the force?

A cubic dice is sliding on a frictionless table. The dice is a cube with edges of length 1 cm and a mass of 30 g. A kid reaches down and gives a horizontal flick to the dice, causing it to change which face is up. What is the minimum impulse in g~cm/s the kid must give to the dice in order to change the face?

Details and assumptions

  • The acceleration of gravity is 9.8 m/s2-9.8~m/s^2.
  • You can model the dice as a perfect cube of uniform density.
  • The dice never leave the table
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