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Torque - Dynamical Behavior

         

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

A fixed pulley-like device is used to lift a box of mass \(m=13\text{ kg},\) as shown in figure above. The outer radius of the device is \( R=7 \text{ m}, \) and the radius of the hub is \(r= 2 \text{ m}. \) When a constant horizontal force \( F = 130 \text{ N} \) 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 \( 2 \text{ m/s}^2. \) Then what is the rotational inertia of the device about its axis of rotation?

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

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

A \( 6 \text{ kg} \) particle that moves on the \(xy\)-plane with velocity \( \vec{v} = \left( 6 \hat{i} + 2 \hat{j} \right) \text{ m/s} \) is at \( x= 9 \text{ m}\) and \(y= 12 \text{ m}. \) It is being pulled by a \( 6 \text{ N} \) force in the \( -x \) 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 \(-9.8~m/s^2\).
  • You can model the dice as a perfect cube of uniform density.
  • The dice never leave the table
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