Biophysics: Friction, Inclines, Translational Motion, and Energy in Physical Activity

Friction

Definition and Properties

Friction is a resistive force that opposes the relative motion or tendency of such motion of two surfaces in contact. It plays a crucial role in everyday phenomena and is essential in understanding motion in physics and biophysics.

• Friction is the resistance to sliding or moving of one surface over another.

• To move an object along a surface, the applied force must overcome the frictional force.

• The coefficient of friction (μ) characterizes the frictional property of surfaces.

• The frictional force also depends on the normal force (Fn) pressing the surfaces together.

Equation for Frictional Force:

• $F_f = \mu F_n$

• Where μ is the coefficient of friction and Fn is the normal force.

Types of Friction

• Kinetic friction ($\mu_k$): Frictional force acting on a moving object.

• Static friction ($\mu_s$): Frictional force acting on an object at rest.

Table: Coefficients of Friction for Various Surfaces

Surface	Static ($\mu_s$)	Kinetic ($\mu_k$)
Leather on oak	0.6	0.5
Rubber on dry concrete	0.9	0.7
Steel on ice	0.02	0.01
Dry bone on bone	0.3	0.3
Bone on joint, lubricated	0.01	0.003

Standing at an Incline

Static Equilibrium on an Inclined Plane

When a person stands on an inclined surface, the forces acting on the body must be analyzed to determine the conditions for sliding.

• Calculate the angle of incline ($\theta$) at which a person of weight W can stand without sliding down.

• Given: $\mu_s = 0.6$ (for oak board).

Relevant Forces:

• Normal force: $F_n = W \cos \theta$

• Static friction force: $F_f = \mu_s F_n = \mu_s W \cos \theta$

• Component of weight parallel to incline: $F_p = W \sin \theta$

Condition for Sliding:

• The person will slide when $F_p \geq F_f$

• $W \sin \theta \geq \mu_s W \cos \theta$

• $\tan \theta = \mu_s$

• For $\mu_s = 0.6$, $\theta = \arctan(0.6) \approx 31^\circ$

Example: A person can stand on an oak board inclined up to $31^\circ$ without sliding.

Friction at the Hip Joint

Biomechanics of Walking

During walking, the forces on the hip joint are significant due to the body's weight and the position of the center of gravity.

• When walking, the full weight of the body rests on one leg during most of each step.

• Because the center of gravity is not directly above the joint, the force on the joint is greater than the body weight.

• Depending on walking speed, this force can be about 2.4 times the weight.

Frictional Wear and Lubrication

• Large forces during motion produce frictional wear at the joints.

• Frictional wear is reduced by cartilage coating and synovial fluid, which lubricate the contact areas.

Application: Understanding joint friction is essential in biomechanics and medical science for preventing joint damage.

Translational Motion

Kinematics of Uniform Acceleration

Translational motion describes the movement of an object from one point to another. The following equations apply to motion with constant acceleration.

• Velocity: $v = v_0 + at$

• Acceleration: $a = \frac{v - v_0}{t}$

• Average velocity: $v_{av} = \frac{v + v_0}{2}$

• Distance traversed: $S = v_0 t + \frac{at^2}{2}$

• Alternative form: $v^2 = v_0^2 + 2aS$

Example: These equations are used to analyze the motion of a person jumping vertically.

Vertical Jump

Forces and Energy in Jumping

Vertical jumping involves the application of forces and the conversion of energy. The analysis includes the forces acting on the jumper and the calculation of acceleration, velocity, and maximum height.

• Two forces act on the jumper: weight (W, downward) and reaction force (F, upward).

• Net upward force: $F - W$

• Acceleration: $a = \frac{F - W}{m} = \frac{F - W}{W/g}$

• Velocity at take-off: $v^2 = v_0^2 + 2ac = 2c \frac{F - W}{W/g}$

• At maximum height $H$: $0 = 2c \frac{F - W}{W/g} - 2gH$

• Maximum height: $H = \frac{(F - W)c}{W}$

Work-Energy Principle:

• Work done on the body equals the potential energy at maximum height: $F c = W (c + H)$

Example: Calculating the height reached in a vertical jump using force and energy equations.

Energy Consumed in Physical Activity

Muscle Work and Efficiency

Physical activity involves the conversion of chemical energy from food into mechanical work and heat. The efficiency of this conversion is typically less than 20%.

• Chemical energy in food is converted to heat and muscle work.

• Muscle efficiency in converting caloric energy to work is less than 20%.

Example Calculation: Energy for Jumping

• Consider a 70 kg person jumping up 60 cm for 10 minutes at a rate of one jump per second.

• Muscle work per jump: $\text{Weight} \times \text{Height} = 70 \times 9.8 \times 0.6 = 411 J$

• Total muscle work in 10 min (600 jumps): $411 J \times 600 = 2.47 \times 10^4 J$

• Energy needed from food (20% efficiency): $\frac{2.47 \times 10^4 J \times 100}{20} = 1.23 \times 10^5 J = 294 \times 10^2 cal = 294 kcal$

Application: This calculation is relevant for understanding energy expenditure in exercise physiology.

Homework Problems

• Problems 2.1, 2.2

• Problems 3.1, 3.2, 3.5

Additional info: These notes cover topics directly relevant to college-level physics, including friction, forces on inclines, translational motion, work, energy, and applications in biophysics.