- #31
smallphi
- 436
- 2
Here is my explanation of the swing problem. Let our system is the man only, without the swing. The only two external forces acting on the man are gravity and the pull of the string of the swing.
Think of what you do intuitively when you want to swing faster. You lower your body to as close to horizontal as possible when the swing is going towards the lowest point (assuming you are facing the lowest point) and again straighten it vertical when the swing passes the lowest point and is going up. You keep your body vertical for half a period till the swing returns to the original position and then you 'pump' again.
Considering the torques of the external forces around the pivot of the swing, only gravity has a torque. By laying horizontal when the swing is going towards its lowest point, you put your center of mass further from the vertical position of the swing increasing the shoulder of the gravity force around the pivot thus adding EXTRA torque of gravity in excess of the usual torque when your body is vertical. That extra torque creates the extra angular acceleration that swings you faster. After you pass the lowest point of the swinging you have to raise your body vertical, otherwise you will have a DEFICIENCY of gravity torque with respect to vertical body position.
That explains how gravity creates the extra angular acceleration. What creates the extra horizontal acceleration though? It can't be gravity since it is vertical. The only other external force is the tension in the string of the swing. What happens is that when gravity creates the extra angular acceleration, the extra speed of the swing requires extra centripetal acceleration provided by EXTRA tension in the string with respect to normally swinging without 'pumping'. That extra tension force projected on the horizontal axis creates the extra momentum in horizontal direction, the extra swing.
Where does the energy for extra swinging come from? Your muscles, say the abdominals and flexors, do work against gravity when you raise your body from horizontal to vertical at the lowest point of the swing. So the chemical energy of the muscles converts to potential energy when you raise your body vertical, which converts to kinetic energy of the swing when you lower down your body horizontal. Of course at the end some part of the chemical muscle energy turns into heat.
Think of what you do intuitively when you want to swing faster. You lower your body to as close to horizontal as possible when the swing is going towards the lowest point (assuming you are facing the lowest point) and again straighten it vertical when the swing passes the lowest point and is going up. You keep your body vertical for half a period till the swing returns to the original position and then you 'pump' again.
Considering the torques of the external forces around the pivot of the swing, only gravity has a torque. By laying horizontal when the swing is going towards its lowest point, you put your center of mass further from the vertical position of the swing increasing the shoulder of the gravity force around the pivot thus adding EXTRA torque of gravity in excess of the usual torque when your body is vertical. That extra torque creates the extra angular acceleration that swings you faster. After you pass the lowest point of the swinging you have to raise your body vertical, otherwise you will have a DEFICIENCY of gravity torque with respect to vertical body position.
That explains how gravity creates the extra angular acceleration. What creates the extra horizontal acceleration though? It can't be gravity since it is vertical. The only other external force is the tension in the string of the swing. What happens is that when gravity creates the extra angular acceleration, the extra speed of the swing requires extra centripetal acceleration provided by EXTRA tension in the string with respect to normally swinging without 'pumping'. That extra tension force projected on the horizontal axis creates the extra momentum in horizontal direction, the extra swing.
Where does the energy for extra swinging come from? Your muscles, say the abdominals and flexors, do work against gravity when you raise your body from horizontal to vertical at the lowest point of the swing. So the chemical energy of the muscles converts to potential energy when you raise your body vertical, which converts to kinetic energy of the swing when you lower down your body horizontal. Of course at the end some part of the chemical muscle energy turns into heat.
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