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Linear Kinetics | |
08/06/04 |
               
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1. Cause of AccelerationAn interaction that causes an acceleration of a body is called a force, which is roughly a push or pull. The relationship between a force and an acceleration which the force causes is known by studying Newtonian mechanics. Even though the Newtonian mechanics is applied to special cases, it is still important to study motion of bodies in Biomechanics. There are three laws in Newtonian mechanics. 2. Newton's First Law of MotionLet's consider a body on which no external force acts. If the body is at rest, it will remain at rest. If the body moving with constant velocity , it will continue to do so. You can imagine that when you are sliding a book on a frictionless surface, then the object will not stop without any external forces. You can also think of a book on a stable desk. It will not move without a force. Newton's first law is also called a law of inertia. Every body continues in its state of rest or motion in a straight line unless compelled to change that state by external forces exerted upon it. 3. Internal, External, and Resultant (Net) ForceInternal forces are forces that are internal to the system and external forces are the forces which are external to the system. The classification of internal and external forces are used for convenience as far as the system is concerned. Internal or external can be changed depending on defining the system in which an object exists. The net force is the vector sum of all forces. 4. MassWe are easily confused by a body's size, weight, or density with a mass. If those are not a mass, and then what is a mass? Mass of a body is the characteristic that relates a force on the body to the resulting acceleration. As you see from the definition of mass, physical sensation of mass can actually be known only when you think of accelerating a body. We will take a look at that in detail from the second law of motion coming up next. 5. Newton's Second Law The
net force
which may be written in its scalar component version
A free-body diagram is useful to deal with the second law. It is a stripped-downdiagram in which only one body is considered. That body is represented by a dot. The external forces on the body are drawn as vectors, and a coordinate system is superimposed, oriented so as to simplify the situation. 6. Some Particular Forces- Weight W : A body's weight W is the force on the body from a nearby astronomical body.
where - Normal force N : A normal force is the force exerted on a body by a surface against which the body is pressed. The normal force is always perpendicular to the surface. Actually the term normal means 'perpendicular'.
- Friction f : A frictional force is the force on a body when the body slides or attempts to slide along a surface. The force is parallel to the surface and directed so as to oppose the motion of the body. - Tension T : A tension is the force on a body from a taut cord at its point of attachment. The force points along the cord. away from the body. For massless cords, the pulls at both ends of the cord have the same magnitude T, even if the cord runs around a massless, frictionless pulley. 7. Newton's Third Law If body A
exerts a force
For every force that is exerted by one body on another there is an equal and opposite force exerted by the second body on the first. 8. Friction- Sliding Friction |
| Property 1 | If the body does not move even while a curtain amount of force F is applied to the body. The magnitude of F is the same as that of Fs and the direction of F is opposite to that of Fs. |
| Property 2 | The maximum static friction Fs, Fs-max is given by
(8.1), where  is
the coefficient of static friction and
is the normal force. |
| Property 3 | The kinetic friction Fk is given by
(8.2), where  is
the coefficient of kinetic frictional force and
is also the normal force. |
Friction Plot
Figure 8.2 Standard friction plot
- Rolling Friction
When a wheel rolls, it requires a certain amount of frictional force, at least, some force which can make the wheel not slip.
A rolling wheel requires a certain amount of friction so that the point of contact of the wheel with the surface will not slip. The amount of traction which can be obtained for an auto tire is determined by the coefficient of static friction between the tire and the road. If the wheel is locked and sliding, the force of friction is determined by the coefficient of kinetic friction and is usually significantly less.
Assuming that a wheel is rolling without slipping, the surface friction does no work against the motion of the wheel and no energy is lost at that point. However, there is some loss of energy and some deceleration from friction for any real wheel, and this is sometimes referred to as rolling friction. It is partly friction at the axle and can be partly due to flexing of the wheel which will dissipate some energy. Figures of 0.02 to 0.06 have been reported as effective coefficients of rolling friction for automobile tires, compared to about 0.8 for the maximum static friction coefficient between the tire and the road.
- Collision
Collision is an event where two or more bodies produce forces on each other for a short time. By knowing the state of bodies before and after the collision, we can figure out mainly the properties of force acting on the bodies. If the bodies are completely elastic, the laws of conservation of momentum and of energy can be applied to the event even though we hardly see this kind of situations In sports.
- Impulse and linear momentum
Impulse is the product of a force F and the time t for which it acts. If the force is variable, the impulse is the integral of Fdt from ti to tf. The impulse of a force acting for a given time interval is equal to the change in momentum produced over that interval. So we can say
J (Impulse) = F t = m a t = m (vf-vi)
(9-1)
(where mass m is a constant and velocity v is a variable)
because the linear momentum p of a body is the product of its mass and its velocity, i.e.
p (momentum) = mv (9-2)
Fig. 9-1
Fig.9-1 shows two bodies, A and B in collision. During rth collision, body A
exerts force F(t) on body B, and body B exerts force -F(t) on body A. Force F(t)
and -F(t) are a pair of action and reaction. The magnitudes of two forces vary
with time and the magnitudes are equal at a given instant.
Theses forces change the linear momentum of the bodies.
From (9-1), (9-2), and the Newton's Second Law, We can have
(9-3)
(9-4)
When we integrate the left and right sides of (9-4), we will get
(9-5)
The left side of (9-5) is pf-pi, which is change in momentum and the
right side is impulse J that is a measure of both the magnitude and duration of
the collision force.
So the impulse is defined as
(9-10)
The equation (9-10) implies that the impulse is equal to the area of F(t) curve in Fig. 9-2.
Fig. 9-2. Impulse graph
The equation (9-10) can be written in component form as
(9-11)
(9-12)
(9-13)
Both momentum and impulse are vectors, so they have both magnitude and direction.
- Elastic Collisions in One Dimension
(1) Stationary Target
When the kinetic energy of the system is the same before and after the collision, we call it elastic collisions. In an elastic collision, the kinetic energy of each colliding body can change, but the total kinetic energy of the system does not change. Therefore we can say
Fig. 9-3
(9-14) Conservation of kinetic energy
(where, m1: mass of moving body, m2: mass of target body initially at rest, v1i: initial velocity of moving body, v1f: final velocity of moving body, and v2f: final velocity of target body)
In a closed and isolated system, the linear momentum of each colliding body can be changed, but the total momentum of the system does not change (Linear Momentum Conservation).
(9-15) Conservation of linear momentum
and also, we can write new equations below from (9-14) and (9-15) - Consider V2i is equal to zero.
(9-16) Final velocity of m1
(9-17) Final velocity of m2
(2) Moving Target
Fig. 9-4
Fig. 9-4 shows the situation in which both bodies are moving (not necessarily moving toward each other). We can rewrite (9-14) and (9-15) for this situation as
(9-18) Final velocity of m1
(9-19) Final velocity of m2
- Inelastic Collision - 1D
An inelastic collision is one in which the kinetic energy of a system
of two colliding bodies is not conserved, but the total linear momentum
of the system is always conserved. When the colliding bodies are stick
together, we call it a completely inelastic collision. In this situation, the
kinetic energy reduction is maximum.
By using the conservation of linear momentum of the system, we can write a
equation for a stationary target situation as
(9-20)
and for a moving target situation as
(9-21)
(where, V is the velocity of both bodies which is stick together)
Figure 8.1 (a) An external force F applied to the block is balanced by the static frictional force Fs until it reaches the maximum static frictional force. In other words, as the F increases, the Fs also increases. (b) The block suddenly break down the balance between F and Fs, and the block moves to the right. (c) When the block moves with a constant velocity, the Fk value is less than the maximum Fs value.
Figure 8.2 Experimental result of frictional force changes while sliding a body (block). (a), (b), and (c) in Figure 8.2 correspond to the (a), (b), and (c) in Figure 8.1
Properties of Friction
| Property 1 | If the body does not move even while a curtain amount of force F is applied to the body. The magnitude of F is the same as that of Fs and the direction of F is opposite to that of Fs. |
| Property 2 | The maximum static friction Fs, Fs-max is given by
(8.1), where  is
the coefficient of static friction and
is the normal force. |
| Property 3 | The kinetic friction Fk is given by
(8.2), where  is
the coefficient of kinetic frictional force and
is also the normal force. |
Friction Plot
Figure 8.2 Standard friction plot
- Rolling Friction
When a wheel rolls, it requires a certain amount of frictional force, at least, some force which can make the wheel not slip.
A rolling wheel requires a certain amount of friction so that the point of contact of the wheel with the surface will not slip. The amount of traction which can be obtained for an auto tire is determined by the coefficient of static friction between the tire and the road. If the wheel is locked and sliding, the force of friction is determined by the coefficient of kinetic friction and is usually significantly less.
Assuming that a wheel is rolling without slipping, the surface friction does no work against the motion of the wheel and no energy is lost at that point. However, there is some loss of energy and some deceleration from friction for any real wheel, and this is sometimes referred to as rolling friction. It is partly friction at the axle and can be partly due to flexing of the wheel which will dissipate some energy. Figures of 0.02 to 0.06 have been reported as effective coefficients of rolling friction for automobile tires, compared to about 0.8 for the maximum static friction coefficient between the tire and the road.
- Collision
Collision is an event where two or more bodies produce forces on each other for a short time. By knowing the state of bodies before and after the collision, we can figure out mainly the properties of force acting on the bodies. If the bodies are completely elastic, the laws of conservation of momentum and of energy can be applied to the event even though we hardly see this kind of situations In sports.
- Impulse and linear momentum
Impulse is the product of a force F and the time t for which it acts. If the force is variable, the impulse is the integral of Fdt from ti to tf. The impulse of a force acting for a given time interval is equal to the change in momentum produced over that interval. So we can say
J (Impulse) = F t = m a t = m (vf-vi)
(9-1)
(where mass m is a constant and velocity v is a variable)
because the linear momentum p of a body is the product of its mass and its velocity, i.e.
p (momentum) = mv (9-2)
Fig. 9-1
Fig.9-1 shows two bodies, A and B in collision. During rth collision, body A
exerts force F(t) on body B, and body B exerts force -F(t) on body A. Force F(t)
and -F(t) are a pair of action and reaction. The magnitudes of two forces vary
with time and the magnitudes are equal at a given instant.
Theses forces change the linear momentum of the bodies.
From (9-1), (9-2), and the Newton's Second Law, We can have
(9-3)
(9-4)
When we integrate the left and right sides of (9-4), we will get
(9-5)
The left side of (9-5) is pf-pi, which is change in momentum and the
right side is impulse J that is a measure of both the magnitude and duration of
the collision force.
So the impulse is defined as
(9-10)
The equation (9-10) implies that the impulse is equal to the area of F(t) curve in Fig. 9-2.
Fig. 9-2. Impulse graph
The equation (9-10) can be written in component form as
(9-11)
(9-12)
(9-13)
Both momentum and impulse are vectors, so they have both magnitude and direction.
- Elastic Collisions in One Dimension
(1) Stationary Target
When the kinetic energy of the system is the same before and after the collision, we call it elastic collisions. In an elastic collision, the kinetic energy of each colliding body can change, but the total kinetic energy of the system does not change. Therefore we can say
Fig. 9-3
(9-14) Conservation of kinetic energy
(where, m1: mass of moving body, m2: mass of target body initially at rest, v1i: initial velocity of moving body, v1f: final velocity of moving body, and v2f: final velocity of target body)
In a closed and isolated system, the linear momentum of each colliding body can be changed, but the total momentum of the system does not change (Linear Momentum Conservation).
(9-15) Conservation of linear momentum
and also, we can write new equations below from (9-14) and (9-15) - Consider V2i is equal to zero.
(9-16) Final velocity of m1
(9-17) Final velocity of m2
(2) Moving Target
Fig. 9-4
Fig. 9-4 shows the situation in which both bodies are moving (not necessarily moving toward each other). We can rewrite (9-14) and (9-15) for this situation as
(9-18) Final velocity of m1
(9-19) Final velocity of m2
- Inelastic Collision - 1D
An inelastic collision is one in which the kinetic energy of a system
of two colliding bodies is not conserved, but the total linear momentum
of the system is always conserved. When the colliding bodies are stick
together, we call it a completely inelastic collision. In this situation, the
kinetic energy reduction is maximum.
By using the conservation of linear momentum of the system, we can write a
equation for a stationary target situation as
(9-20)
and for a moving target situation as
(9-21)
(where, V is the velocity of both bodies which is stick together)
This site was last updated 10/18/03
on a body
with mass m is related to the body's acceleration a by 
(5.1) 
,
(5.2).
The second law indicates that in SI units
(5.3). 
(6.1)
is the free-fall acceleration.
Usually the astronomical body is the Earth in Biomechanics.
Figure.6.1
on body B, then B must
exert a force 
on body A. The forces are
equal in magnitude and opposite in direction:
(7.1)