Basic Biomechanics, (5th edition) by Susan J. Hall, Ph.D. Chapter 12 Linear Kinetics of Human Movement

1st Newton’s Laws What is the law of inertia? A body will maintain a state of rest or constant velocity unless acted on by an external force that changes the state.

2nd Newton’s Laws What is the law of acceleration? A force applied to a body causes acceleration of that body of a magnitude proportional to the force in the direction of the force and inversely proportional to the body’s mass F = ma

3rd Newton’s Laws What is the law of reaction? For every action, there is an equal and opposite reaction. When one body exerts a force on a second, the second body exerts a reaction force that is equal in magnitude and opposite in direction on the first body.

Newton’s Laws The weight of a box sitting on a table generates a reaction force by the table that is equal in magnitude and opposite in direction to the weight. wt R

Mechanical Behavior of Bodies in Contact What is friction? A force acting over the area of contact between two surfaces direction is opposite of motion or motion tendency magnitude is the product of the coefficient of friction () and the normal reaction force (R); F = R

Mechanical Behavior of Bodies in Contact Static Fm = sR Dynamic Fk = kR Applied external force Friction For static bodies, friction is equal to the applied force. For bodies in motion, friction is constant and less than maximum static friction.

Example: The coefficient of static friction between a sled and the snow is 0.18, with a coefficient of kinetic friction of 0.15. A 250 N boy sits on the 200 N sled. How much force directed parallel to the horizontal surface is required to start the sled in motion. How much force is required to keep the sled in motion?

Mechanical Behavior of Bodies in Contact Pushing a desk Pulling a desk R = wt + Pv R = wt - Pv wt Is it easier to push or pull a desk across a room?

Example: If μs between a basketball shoe and a court is 0.56, and the normal reaction force acting on the shoe is 350 N, how much horizontal force is required to cause the shoe to slide?

Solution: μs = 0.56, R = 350 N Fm = μsR = 0.56(350) = 196 N (greater than)

Mechanical Behavior of Bodies in Contact What is momentum? quantity of motion possessed by a body measured as the product of a body’s mass and its velocity; M = mv

Mechanical Behavior of Bodies in Contact What is the principle of conservation of momentum? In the absence of external forces, the total momentum of a given system remains constant. M1 = M2 (mv)1 = (mv)2

Example: Two skaters gliding on ice run into each other head-on. If the two skaters hold onto each other and continue to move as a unit after the collision, what will be their resultant velocity? Skater A has a velocity of 5 m/s and a mass of 65 kg. Skater B has a velocity of 6 m/s and a mass of 60 kg.

Solution: ma = 65 kg, va = 5 m/s, mb = 60 kg, vb = -6 m/s m1v1 + m2v2 = (m1 + m2)v 65(5) + 60(-6) = (65 + 60)v v = -35/125 v = 0.28 m/s B pushing A

Mechanical Behavior of Bodies in Contact What causes momentum? impulse: the product of a force and the time interval over which the force acts

Mechanical Behavior of Bodies in Contact What is the relationship between impulse and momentum? Ft = M Ft = (mv)2 - (mv)1

Mechanical Behavior of Bodies in Contact What does the area under the curve represent? 3 2 1 Time (ms) 50 100 150 200 250 A 50 100 150 200 250 B Force (Body Weight) Force-time graphs from a force platform for high (A) and low (B) vertical jumps by the same performer.

Example: A baseball with an initial velocity of 20 ms-1 is hit by a player and then moving in the opposite direction. After the ball is hit, it moves with a velocity of 36 ms-1. The ball has a mass of 0.16 kg and the time of impact is 8.0 x 10-3 s. Calculate: The impulse applied to the ball The impulse force exerted on the ball by the bat

Mechanical Behavior of Bodies in Contact What is impact? a collision characterized by: the exchange of a large force during a small time interval

Mechanical Behavior of Bodies in Contact What happens following an impact? This depends on: the momentum present in the system the nature of the impact

Mechanical Behavior of Bodies in Contact What is the nature of impact? It is described by the coefficient of restitution, a number that serves as an index of elasticity for colliding bodies; represented as e.

Mechanical Behavior of Bodies in Contact What does the coefficient of restitution describe? relative velocity after impact -e = relative velocity before impact v1 - v2 -e = u1 - u2

Mechanical Behavior of Bodies in Contact Ball velocities before impact Ball velocities after impact u1 u2 v1 v2 v1 - v2 = -e ( u1 - u2) The differences in two balls’ velocities before impact is proportional to the difference in their velocities after impact. The factor of proportionality is the coefficient of restitution.

Mechanical Behavior of Bodies in Contact The coefficient of restitution between a ball and a flat, stationary surface onto which the ball is dropped may be approximated using the following formula: e = √(hb/hd) hb = the height to which the ball bounces hd = the height from which the ball is dropped

Example: A ball dropped on a surface from a 2 m height bounces to a height of 0.98 m. What is the coefficient of restitution between ball and surface?

Solution: e = √(hb/hd) e = √(0.98/2) e = 0.7

Mechanical Behavior of Bodies in Contact What kinds of impact are there? perfectly elastic impact - in which the velocity of the system is conserved; (e = 1) perfectly plastic impact - in which there is a total loss of system velocity; (e = 0) (Most impacts fall in between perfectly elastic and perfectly plastic.)

Work, Power, and Energy Relationships What is mechanical work? the product of a force applied against a resistance and the displacement of the resistance in the direction of the force W = Fd units of work are Joules (J)

Work, Power, and Energy Relationships What is mechanical power? the rate of work production calculated as work divided by the time over which the work was done W t units of work are Watts (W) P =

Example: A set of 20 stairs, each of 20 cm height, is ascended by a 700 N man in a period of 1.25 seconds. Calculate the mechanical work, power, and change in potential energy during the ascent.

Solution: W = Fd, P = W/t, PE = wt(h) F = 700 N d or h = 20 X 20 cm = 400 cm = 4m t = 1.25 s W = 700(400) = 2800 Nm = 2800 J P = 2800/1.25 = 2240 J/s = 2240 W PE = 700(4) = 2800 J

Work, Power, and Energy Relationships What is mechanical energy? the capacity to do work units of energy are Joules (J) there are three forms energy: kinetic energy potential energy thermal energy

Work, Power, and Energy Relationships What is kinetic energy? energy of motion KE = ½mv2 What is potential energy? energy by virtue of a body’s position or configuration PE = (wt)(h)

Work, Power, and Energy Relationships What is the law of conservation of mechanical energy? When gravity is the only acting external force, a body’s mechanical energy remains constant. KE + PE = C (where C is a constant - a number that remains unchanged)

Work, Power, and Energy Relationships 29.4 24.5 19.6 14.7 9.8 3.1 4.4 5.4 6.3 4.9 3.0 2.5 2.0 1.5 1.0 Ht(m) PE(J) V(m/s) KE(J) Time Height, velocity, potential energy, and kinetic energy changes for a tossed ball. Note: PE + KE = C

Example: Using the principle of conservation of mechanical energy, calculate the maximum height achieved by a 7 N ball tossed vertically upward with an initial velocity of 10 m/s.

Work, Power, and Energy Relationships What is the principle of work and energy? The work of a force is equal to the change in energy that it produces in the object acted upon. W = KE + PE + TE (where TE is thermal energy)

Linear Kinetics of Human Movement Chapter 12 Linear Kinetics of Human Movement