Question
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Close a door by pushing on the doorknob. Close the door again using approximately the same force but applying it 5 centimeters away from the hinges. Try again by applying the force on the edge of the door in a direction pointing toward the hinges. Summarize and explain your observations. Comment on the difference between force and torque. Newton's second law could be stated either as F- = ma-, or as force equals time rate of change of linear momentum. For simple rotational motion the corresponding law is stated as tau = l alpha-, or torque equals time rate of change of angular momentum. Momentum is conserved when there is no external force. Angular momentum is conserved when there is no external torque. To test this, sit on a rotating stool with your hands in by your sides. While rotating, extend your arms away from your body. Repeat while holding masses in your hands. Describe the effects. Give an explanation. A heavy mass in free fall does not fall faster than a light mass. The extra weight is compensated for by extra resistance. A heavy mass on a frictionless plane does not slide down faster than a light mass. A heavy ring and a light ring both roll down an inclined plane with the same speed. What about a ring and a disk? Try this with combinations of rings and disks of different masses. Describe the results and give an explanation. It may be helpful to write down an energy equation for the case of 2 rings of different radii and different masses. Also try the energy equation for a ring and a disk of the same radius and mass. This demonstration is more complicated because it involves a vector. For a rotating wheel, the direction of the angular momentum, L-, is found by curving the fingers of your right hand in the direction of the rotation, your thumb then points along L-. The angular momentum vector is perpendicular to the plane of the wheel. Sit on the rotating stool used in part II. Hold a bicycle wheel with its axis vertical. While the stool is at rest, have your partner spin the bicycle wheel. Carefully note the direction of spin of the wheel and of the stool. Give an explanation in terms of conservation of angular momentum. While the stool is still rotating, carefully turn the bicycle wheel upside down. Explain what happens. This is an interesting phenomenon, and a very difficult one to explain. Both torque and angular momentum must be treated as vectors. Torque causes angular momentum to change according to the following equation: delta L = tau delta t
Explanation / Answer
a) Torque is needed to close the door.
Torque = Force*distance
For same torque, when distance is more, force required is less.
Hence, when we apply force at 5 cm away from hinges, the force required would be more compared to when force is applied at the door which is much farther from the hinges.
When force is applied pointing towards the hinges, Torque is zero since the "perpendicular" distance between line of action of force and door axis is zero.
2.
Torque = Inertia*angular acceleration
Angular momentum = Inertia*anglar velocity = constant (in absence of external torque)
When we extend our hands, more mass of body is away from the rotation axis and thereby our inertia increases. Since angular momentum is constant, the angular velocity will reduce.
Holding masses in hands will furhter increase the inertia which will further reduce the angular velocity.
3.
Energy conservation:
PE = KE
mgh = 1/2*mv^2 + 1/2*Iw^2
w = v/r
So, mgh = 1/2*mv^2 + 1/2*Iv^2/r^2
mgh = 1/2*v^2 [m + I/r^2]
Thus, for a given mass and initial height, the velocity at the bottom of the incline for the shape for which I/r^2 is lower.
For ring, I = mr^2 and for disk I = 1/2*mr^2
Snce, I/r^2 is lower for disk, its velocity at bottom of incline plane would be higher than ring.
4.
When stool is rotating clockwise, wheel will rotate counter-clockwise (and vice versa) since angular momentum of stool+wheel remains conserved.
