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Torque

From twisting the lid off a jar of olives, to balancing the tandem bicycle you're riding with your parole officer, torque explains it all. Learn to describe and calculate torque, the "twisting force".

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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