DP IB Physics SL

A. Space, time and motion

A.2 Forces and momentum

DP IB Physics: SL A. Space, time and motion A.2 Forces and momentum Understandings Students should understand:
a)	Newton’s three laws of motion
b)	Forces as interactions between bodies
c)	That forces acting on a body can be represented in a free-body diagram
d)	that free-body diagrams can be analyzed to find the resultant force on a system
e)	The nature and use of the following contact forces – Normal force [math]F_N[/math] is the component of the contact force acting perpendicular to the surface that counteracts the bod – Surface frictional force [math]F_f[/math] acting in a direction parallel to the plane of contact between a body and a surface, on a stationary body as given by [math]F_f ≤ μ_s F_N[/math] or a body in motion as given by [math]F_f = μdFNn [/math] where [math]μ_s[/math] and [math]μ_d[/math] are the coefficients of static and dynamic friction respectively – Tension – Elastic restoring force [math]F_H[/math] following Hooke’s law as given by [math]F_H = –kx[/math] where k is the spring constant – Viscous drag force [math]F_d[/math] acting on a small sphere opposing its motion through a fluid as given by [math]F_d = 6πηrv[/math] where η is the fluid viscosity, r is the radius of the sphere and v is the velocity of the sphere through the fluid – Buoyancy [math]F_b[/math] acting on a body due to the displacement of the fluid as given by [math]F_b = ρVg<[/math] where V is the volume of fluid displaced
f)	The nature and use of the following field forces – Gravitational force [math]F_g[/math] is the weight of the body and calculated is given by [math]F_g = mg[/math] – Electric force [math]F_e[/math] – Magnetic force [math]F_m[/math]
g)	That linear momentum as given by [math]p = mv[/math] remains constant unless the system is acted upon by a resultant external force
h)	That a resultant external force applied to a system constitutes an impulse [math]J[/math] as given by [math]J = FΔt[/math] where F is the average resultant force and [math]Δt[/math] is the time of contact
i)	That the applied external impulse equals the change in momentum of the system
j)	That Newton’s second law in the form [math]F = ma[/math] assumes mass is constant whereas [math]F = \frac{∆p}{∆t}[/math] allows for situations where mass is changing
k)	The elastic and inelastic collisions of two bodies
l)	Explosions
m)	Energy considerations in elastic collisions, inelastic collisions, and explosions
n)	That bodies moving along a circular trajectory at a constant speed experience an acceleration that is directed radially towards the center of the circle—known as a centripetal acceleration as given by [math]a = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2 r}{T^2}[/math]
o)	That circular motion is caused by a centripetal force acting perpendicular to the velocity
p)	That a centripetal force causes the body to change direction even if its magnitude of velocity may remain constant
q)	That the motion along a circular trajectory can be described in terms of the angular velocity ω which is related to the linear speed v by the equation as given by [math]v = \frac{2\pi r}{T} = \omega r[/math]

a. Newton’s Three Laws of Motion
Sir Isaac Newton formulated three fundamental laws of motion, which describe the relationship between force, motion, and mass. These laws form the basis of classical mechanics.
⇒ 1st Law: Law of Inertia
“An object at rest remains at rest, and an object in motion continues in motion with the same velocity unless acted upon by an external force.”
Figure 1 Law of Inertia
Explanation:
If no force acts on an object, it will stay still or move with constant velocity.
A force is required to change the motion (start, stop, or alter direction).
This property of an object to resist changes in its motion is called inertia.
Example:
A passenger in a car jerk forward when the brakes are suddenly applied because their body wants to remain in motion.
⇒ 2nd Law of motion:
“The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.”
Figure 2 2^nd law of motion
Mathematically,
[math]F = ma[/math]
Where:
– F = Force (N)
– m = Mass (kg)
– a = Acceleration (m/s²)
Explanation:
– A greater force produces greater acceleration.
– A larger mass resists acceleration more (heavier objects require more force to move).
Example:
A car with a greater mass needs more force to accelerate than a bicycle.
Example Calculation:
A 5 kg box is pushed with 20 N of force. What is its acceleration?
[math]F = ma \\
a = \frac{F}{m} \\
a = \frac{20}{5} \\
a = 4\, \text{m/s}^2[/math]
⇒ 3rd Law: Action-Reaction
“For every action, there is an equal and opposite reaction.”
Figure 3 3^rd Law of motion
Explanation:
Forces always come in pairs—when one body exerts a force on another, the second body exerts an equal force in the opposite direction.
Example:
When you jump, your feet push down on the ground, and the ground pushes you upward with an equal force.
A rocket moves upward because it expels gas downward.
b. Forces as Interactions Between Bodies
⇒ Types of Forces
Forces can be classified as:
1. Contact Forces
– Friction (resists motion)
– Tension (force in a rope or cable)
– Normal Force (reaction from a surface)
– Applied Force (push/pull)
2. Non-Contact Forces
– Gravitational Force (weight)
– Electromagnetic Force (electric & magnetic forces)
– Nuclear Forces (strong and weak forces in atoms)
Examples of Force Interactions
A book on a table is pulled downward by gravity, but the table pushes back (normal force).
A magnet attracts a metal object due to magnetic force.
Figure 4 Force as interaction between bodies
c. Free-Body Diagrams (FBDs)
A free-body diagram represents all the forces acting on an object.
How to Draw a Free-Body Diagram
1. Draw the object as a point or box.
2. Identify all forces acting on it.
3. Draw arrows starting from the object’s center, showing the direction of each force.
4. Label the forces (e.g., [math]F_g[/math] for gravity, [math]F_N[/math] for normal force).
Figure 5 Free body diagram
⇒ Example: A Box on a Table
– Gravity ([math]F_g[/math]) pulls downward.
– Normal Force ([math]F_N[/math]) from the table pushes up.
– If someone pushes the box horizontally, there’s an applied force ([math]F_A[/math]).
– Friction ([math]F_f[/math]) opposes motion.
⇒ Example: A Falling Parachutist
1. Weight ([math]F_g[/math]) pulls them down.
2. Air resistance ([math]F_D[/math]) opposes motion upwards.
3. As speed increases, air resistance increases until it balances weight → terminal velocity.
d. Analyzing Free-Body Diagrams to Find the Resultant Force
A free-body diagram (FBD) is a simplified representation of an object showing all the forces acting on it. It helps analyze motion and determine the resultant force.
The resultant force ([math]F_{net}[/math]) is the vector sum of all forces acting on an object. It determines:
– Whether the object is at rest or in motion.
– The direction and magnitude of acceleration (if any).
– It is calculated using vector addition.
Figure 6 Free body diagram to find the resultant force
⇒ Newton’s Second Law:
[math]F_{net} = ma[/math]
Where:
– [math]F_{net}[/math] = Resultant force (N)
– m = Mass of the object (kg)
– a = Acceleration (m/s²)
⇒ Steps to Analyze a Free-Body Diagram
1. Identify the Object
Choose a single object to analyze.
2. Identify All Forces
Draw arrows from the object’s center, showing all applied forces.
Label forces (e.g., [math]F_g[/math] for gravity, [math]F_N[/math] for normal force).
3. Resolve Forces into Components
Break diagonal forces into horizontal (x) and vertical (y) components using trigonometry.
4. Find the Net Force in Each Direction
Sum forces along the horizontal axis ([math]F_{net}. x[/math]).
Sum forces along the vertical axis ([math]F_{net} . y[/math]).
5. Calculate the Resultant Force
Use Pythagoras’ theorem to find magnitude:
[math]F_{\text{net}} = \sqrt{(F_{\text{net},x})^2 + (F_{\text{net},y})^2}[/math]
Use trigonometry to find direction:
[math]\theta = \tan^{-1}\left( \frac{F_{\text{net},y}}{F_{\text{net},x}} \right)[/math]
Example 1: A Box on a Table with a Pushing Force
⇒ Given Forces:
1. Gravity ([math]F_g[/math]): 50 N downward.
2. Normal Force ([math]F_N[/math]): 50 N upward.
3. Applied Force ([math]F_A[/math]): 20 N to the right.
4. Friction ([math]F_f[/math]): 5 N to the left.
Step 1: Sum Forces in Each Direction
Vertical forces:
[math]F_N – F_g = 50 – 50 = 0 → \text{No vertical motion}[/math]
Horizontal forces:
[math]F_A – F_f = 20 – 5 = 15 N \text{ to right}[/math]
Step 2: Find the Resultant Force
Since the vertical forces cancel, the resultant force is 15 N to the right.
e. Nature and Use of Contact Forces
Contact forces arise when two objects physically interact. These forces can act perpendicular (normal force) or parallel (friction) to the surface of contact. Other contact forces include tension, elastic restoring force, viscous drag, and buoyancy.
1. Normal Force ([math]F_N[/math])
The normal force is the contact force exerted by a surface perpendicular to an object resting on it. It counteracts gravity and prevents objects from falling through solid surfaces.

Figure 7 Normal force
⇒ Formula for Normal Force:
If an object of mass mmm is resting on a horizontal surface:
[math]F_N = mg[/math]
If the object is on an inclined plane at angle
[math]F_N = mg cos⁡θ[/math]
⇒ Examples of Normal Force:
A book resting on a table experiences an upward normal force that balances its weight.
A person standing on the ground feels a normal force equal to their weight.
2. Surface Frictional Force ([math]F_f[/math])
Friction is a contact force that resists the motion of an object parallel to the contact surface
⇒ Types of Friction:
– Static Friction ([math]F_s[/math]) – Prevents an object from starting to move.
[math]F_s ≤ μ_s F_N[/math]
Where is the coefficient of static friction.
– Dynamic (Kinetic) Friction ([math]F_d[/math]) – Acts when an object is already moving.
[math]F_d = μ_d F_N[/math]
Where is the coefficient of kinetic friction ([math]μ_d < μ_s[/math]).
Figure 8 Frictional forces
⇒ Examples of Friction:
A car tire gripping the road depends on static friction.
A moving object sliding on ice experiences kinetic friction, which slows it down.
– Tension (T)
Tension is the pulling force transmitted by a string, rope, or cable when stretched by forces acting from opposite ends.

Figure 9 Tension force apply on a string
⇒ Formula for Tension:
If a mass mmm is hanging from a rope:
[math]T = mg[/math]
If an object is accelerating upward with acceleration a:
[math]T = m(g + a)[/math]
If an object is accelerating downward:
[math]T = m(g – a)[/math]
⇒ Examples of Tension:
A person holding a hanging mass experiences tension in the rope.
A bridge suspension cable holds the weight of the bridge by tension forces.
– Elastic Restoring Force ([math]F_H[/math]) – Hooke’s Law
A stretched or compressed elastic material (like a spring) exerts a force that restores it to its original length. This follows Hooke’s Law:
[math]F_H – kx[/math]
Where:
k = Spring constant (N/m), a measure of stiffness.
x = Displacement from the equilibrium position (m).
⇒ Examples of Elastic Force:
A stretched rubber band exerts a restoring force when released.
Shock absorbers in vehicles use springs to reduce impact force.
– Viscous Drag Force ([math]F_d[/math]) – Stokes’ Law
Drag force opposes the motion of an object moving through a fluid (liquid or gas). For a small sphere moving slowly through a fluid, the Stokes’ Law equation applies:
[math]F_d = 6πηrv[/math]
Where:
η = Fluid viscosity (Pa·s).
r = Radius of the sphere (m).
v = Velocity of the sphere (m/s).
⇒ Examples of Viscous Drag:
A rain droplet falling experiences air resistance, slowing its descent.
A submarine moving through water faces drag due to fluid resistance.
– Buoyancy Force ([math]F_b[/math]) – Archimedes’ Principle
An object submerged in a fluid experience an upward buoyant force equal to the weight of the displaced fluid.
[math]F_b = ρVg[/math]
Where:
ρ = Fluid density (kg/m³).
V = Volume of displaced fluid (m³).
g = Acceleration due to gravity (9.81 m/s²).
Examples of Buoyancy:
A boat floats because the buoyant force balances its weight.
A helium balloon rises since helium is less dense than air.
f. Field forces and momentum principles:
Field forces are non-contact forces that act on objects at a distance due to the presence of a force field, such as gravitational, electric, and magnetic forces. Momentum and impulse describe how objects respond to these forces.
– Gravitational Force (F_g) – Weight of a Body
Nature of Gravitational Force:
Gravitational force is the attraction between two masses.
Every object with mass experiences weight due to Earth’s gravity.
The gravitational force on an object near Earth’s surface is called weight.
Figure 10 Linear momentum and impulse
Formula for Weight:
[math]F_g = mg[/math]
Where:
[math]F_g[/math] = Gravitational force (N).
m = Mass of the object (kg).
g = Acceleration due to gravity (9.81 m/s² on Earth).
⇒ Examples of Gravitational Force:
A dropped ball falls due to gravity.
Planets orbit the Sun due to gravitational attraction.
– Electric Force ([math]F_e[/math]) – Coulomb’s Law
Nature of Electric Force:
Electric forces arise due to charged objects.
Like charges repel, and opposite charges attract.
The force between two charges follows Coulomb’s Law.
Formula for Electric Force:
[math]F_e = \frac{k q_1 q_2}{r^2}[/math]
Where:
k = Coulomb’s constant ([math]9.0 \times 10^9\ \text{N}\,\text{m}^2/\text{C}^2[/math]).
[math]q_1, q_2[/math] = Magnitudes of the charges (C).
r = Distance between the charges (m).
⇒ Examples of Electric Force:
A rubber balloon rubbed on hair sticks to a wall due to electric charge.
Lightning occurs due to electric charge build-up in clouds.
– Magnetic Force ( [math]F_m[/math]) – Lorentz Force
⇒ Nature of Magnetic Force:
Moving charges produce a magnetic field.
A charged particle moving in a magnetic field experiences a force.
Formula for Magnetic Force on a Moving Charge:
[math]F_m = qvBsinθ[/math]
Where:
[math]F_m[/math] = Magnetic force (N).
q = Charge of the particle (C).
v = Velocity of the charge (m/s).
B = Magnetic field strength (T).
θ = Angle between velocity and the field.
⇒ Examples of Magnetic Force:
A compass needle aligns with Earth’s magnetic field.
Electric motors work by generating a force on current-carrying wires in a magnetic field.

g. Linear Momentum ([math]p = mv[/math]) – Conservation Law
⇒ Nature of Linear Momentum:
Momentum measures an object’s tendency to stay in motion.
It is conserved in the absence of external forces.
Formula for Momentum:
[math]p = mv[/math]
Where:
p = Linear momentum ([math]kg. m/s[/math]).
m = Mass of the object (kg).
v = Velocity of the object (m/s).
Figure 11 Conservation of momentum
⇒ Conservation of Momentum:
If no external force acts on a system, the total momentum remains constant.
Newton’s Third Law explains this: for every action, there is an equal and opposite reaction.
⇒ Examples of Momentum Conservation:
A bullet fired from a gun causes recoil due to momentum conservation.
A rocket propels forward by expelling gas in the opposite direction.
h. Impulse (J) – Force and Time Relationship
⇒ Nature of Impulse:
Impulse changes an object’s momentum.
It depends on force and time of interaction.
Increasing the time of impact reduces force, e.g., airbags slow down motion to reduce injury.
Formula for Impulse:
[math]J = FΔt[/math]
Where:
J = Impulse (Ns).
F = Average resultant force (N).
Δt = Time interval (s).
Figure 12 Momentum and impulse
⇒ Impulse-Momentum Theorem:
[math]J = Δp = mv_f – mv_i[/math]
This means impulse equals the change in momentum.
Examples of Impulse:
A cricketer moving their hands backward while catching a ball reduces force on their hands
Seatbelts increase stopping time, reducing impact force on passengers.
i. Momentum, impulse, and collisions:
Momentum and impulse play a crucial role in understanding motion, especially in collisions, explosions, and varying mass systems. These concepts are directly linked to Newton’s Second Law.
1. Impulse and Change in Momentum
Nature of Impulse and Momentum Relationship
Impulse is the effect of a force acting over a time interval.
Momentum measures the quantity of motion of a body.
The impulse-momentum theorem states that the impulse applied to an object is equal to its change in momentum.
Figure 13 Momentum impulse and collusion
Mathematical Form of Impulse-Momentum Theorem
[math]J = \Delta p \\
J = m v_f – m v_i \\
J = F \Delta t[/math]
Where:
– J = Impulse (Ns).
– p = Momentum (kg. m/s).
– m = Mass of the object (kg).
– [math]v_f[/math] = Final velocity (m/s).
– [math]v_i[/math]= Initial velocity (m/s).
– F = Force applied (N).
– [math]\Delta t[/math] = Time of interaction (s).
A large force acting for a short time or a small force acting for a long time produces the same impulse.
⇒ Examples
A tennis racket hitting a ball applies impulse to change its speed.
Airbags in cars increase stopping time, reducing the force on passengers.
j. Newton’s Second Law and Changing Mass
Newton’s Second Law in Standard Form ([math]F = ma[/math])
– This assumes mass is constant, and only acceleration changes.
Newton’s Second Law in Terms of Momentum ([math]F = \frac{\Delta p}{\Delta t}[/math] )
– When mass is changing, such as in rockets, Newton’s Second Law is written as:
[math]F = \frac{\Delta p}{\Delta t}[/math]
– This equation accounts for cases where both mass and velocity
⇒ Examples of Changing Mass Systems
Rocket propulsion: A rocket ejects fuel, decreasing its mass while increasing velocity.
Water jets: Fire hoses eject water, transferring momentum to firefighters.
k. Elastic and Inelastic Collisions
⇒ Types of Collisions
1. Elastic Collision:
– Both momentum and kinetic energy are conserved.
– No loss of energy as heat or sound.
– Example: Two billiard balls colliding.
Figure 14 Elastic collision
2. Inelastic Collision:
– Momentum is conserved, but kinetic energy is lost.
– Some energy is converted into heat, sound, or deformation.
– Example: A car crash where the vehicles stick together.
Figure 15 Inelastic collision
⇒ Equations for Collisions
Momentum conservation (for all collisions):
[math]m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2[/math]
Where
– [math]m_1, m_2[/math] = masses of objects (kg).
– [math]u_1, u_2[/math] = initial velocities (m/s).
– [math]v_1, v_2[/math] = final velocities (m/s).
Elastic Collisions – Kinetic Energy Conservation:
[math]\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2[/math]
Perfectly Inelastic Collisions – Objects Stick Together:
[math]m_1 u_1 + m_2 u_2 = (m_1 + m_2 )v[/math]
⇒ Examples of Collisions
– Elastic: Gas molecules bouncing off each other in air.
– Inelastic: A clay ball hitting the ground and not bouncing.
l. Explosions and Momentum Conservation
⇒ Nature of Explosions
An explosion is the reverse of a collision:
– A system starts at rest ([math]p = 0[/math]) and then fragments move apart.
– Momentum is conserved in explosions.
⇒ Equation for Explosions
[math]m_1 u_1 + m_2 u_2 = 0[/math]
The momenta of the fragments add to zero, meaning they move in opposite directions.
⇒ Examples of Explosions
– A gun recoiling after firing a bullet.
– A firework exploding, sending pieces outward.
m. Circular motion, collisions, and explosions:
1. Energy Considerations in Collisions and Explosions
Collisions and explosions involve the principles of momentum and energy conservation. The nature of energy transfer distinguishes elastic and inelastic
⇒ Elastic Collisions
Momentum is conserved:
[math]m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2[/math]
Kinetic energy is also conserved:
[math]\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_1 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_1 v_2^2/math]
No energy is lost as heat or sound.
Example: Two billiard balls colliding and bouncing apart.
⇒ Inelastic Collisions
Momentum is conserved:
[math]m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2[/math]
Kinetic energy is not conserved because some energy is converted into heat, sound, or deformation:
[math]\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_1 u_2^2 > \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_1 v_2^2[/math]
Example: A car crash where vehicles crumple.
⇒ Explosions
Momentum before explosion = 0, so the sum of momentum of the fragments remains zero:
[math]m_1 u_1 + m_2 u_2 = 0[/math]
Kinetic energy increases as chemical, nuclear, or stored energy is released.
Example: A gun recoiling when a bullet is fired.
n. Centripetal Acceleration in Circular Motion
An object moving in a circular path at constant speed is accelerating because its direction is constantly changing.
Centripetal Acceleration Formula
[math]a = \frac{v^2}{r} \\
a = \omega^2 r \\
a = \frac{4\pi^2 r}{T^2}[/math]
Where:
– a = Centripetal acceleration (m/s²)
– v = Linear speed (m/s)
– r = Radius of circular path (m)
– ω = Angular velocity (rad/s)
– T = Time period for one full rotation (s)
Figure 16 Uniform circular motion
Why Does This Acceleration Exist?
The object moves in a circular path, meaning its velocity direction keeps changing.
Since acceleration is change in velocity, the object is always accelerating toward the center.
This acceleration is called centripetal acceleration.
⇒ Non-Uniform circular motion:
Non-uniform circular motion occurs when an object moves in a circular path with a varying speed. Unlike uniform circular motion (where speed remains constant), non-uniform motion includes both:
Tangential acceleration ([math]a_t[/math]): Due to a change in speed along the circular path.
Centripetal acceleration ([math]a_c[/math]): Due to a change in direction, always pointing toward the center.
Thus, the total acceleration is a combination of these two components.
⇒ Components of Acceleration in Non-Uniform Circular Motion
The total acceleration (a) has two perpendicular components:
Figure 17 non-Uniform circular motion
The weight of a body and centripetal force
[math]F = mg \\
F = \frac{mv^2}{r} \\
mg = \frac{mv^2}{r} \\
g = \frac{v^2}{r} \\
v = \sqrt{gr} \\
v = \sqrt{(9.81)(0.5)} \\
v = 2.21\ \text{m/s}[/math]
(i) Radial (Centripetal) Acceleration ([math]a_c[/math])
[math]a_c = \frac{v^2}{r}[/math]
– Points toward the center of the circle.
– Causes the object to change direction but does not affect speed.
(ii) Tangential Acceleration ([math]a_t[/math])
[math]a_t = \frac{dv}{dt}[/math]
– Acts along the direction of motion (tangent to the circle).
– Increases speed if in the direction of velocity and decreases speed if opposite to velocity.
(iii) Total Acceleration (a)
Since [math]a_c[/math] and [math]a_t[/math] are perpendicular, the magnitude of total acceleration is found using Pythagoras:
[math]a = \sqrt{a_c^2 + a_t^2}[/math]
And the angle θ of acceleration with respect to the radius is:
[math]\tan \theta = \frac{a_t}{a_c}[/math]
Where θ is measured from the radial direction toward the tangential direction.
o. Centripetal Force and Its Role in Circular Motion
Centripetal force is always directed toward the center.
It is perpendicular to the velocity.
It causes the object to change direction but not magnitude of speed.
⇒ Equation for Centripetal Force
[math]F = \frac{mv^2}{r} \\
F = m \omega^2 r[/math]
Where:
– F = Centripetal force (N)
– m = Mass of the object (kg)
– v = Linear speed (m/s)
– r = Radius of circular path (m)
– ω = Angular velocity (rad/s)
⇒ Examples of Centripetal Force
Car moving around a curve: Friction provides the centripetal force.
Satellite orbiting Earth: Gravity provides the centripetal force.
Swinging a ball on a string: Tension in the string provides the centripetal force.
p. Centripetal Force and Change in Direction
A centripetal force is a force that acts toward the center of a circular path, keeping an object in uniform circular motion. Even though the object moves at a constant speed, the presence of this force continuously changes its direction, which means the object is always accelerating.
1. The Object Change Direction:
Velocity has both magnitude and direction. Even if the speed remains constant, a change in direction means velocity is changing.
Acceleration is the rate of change of velocity. Since the velocity is changing (due to direction), the object is accelerating toward the center.
Newton’s First Law states that an object in motion will continue in a straight line unless acted upon by an external force. The centripetal force prevents straight-line motion and forces the object into a circular path.
2. Example Scenarios
Car turning around a curve:
Friction between the tires and road provides the centripetal force, pulling the car toward the center of the turn.
Swinging a ball on a string:
The tension in the string pulls the ball toward the center, changing its direction.
Earth orbiting the Sun:
Gravity acts as a centripetal force, keeping Earth in its elliptical orbit instead of flying off in a straight line.
3. Centripetal Force Formula
The force required to keep an object moving in a circle is:
[math]F = \frac{mv^2}{r} \\
F = m \omega^2 r[/math]
Where:
– F = Centripetal force (N)
– m = Mass of object (kg)
– v = Linear velocity (m/s)
– r = Radius of circular path (m)
– ω = Angular velocity (rad/s)
⇒ Example:
Banking:
When a vehicle moves along a curved path, it requires a centripetal force to keep it moving in the curved trajectory. On a level road, this force is provided solely by friction between the tires and the road.
However, if the road is banked (inclined at an angle), the normal reaction force from the surface also contributes to the required centripetal force, reducing reliance on friction.
⇒ Forces Acting on a Banked Curve
Consider a vehicle of mass m moving at speed v along a curved road that is banked at an angle θ with a radius of curvature r. The forces acting on the vehicle are:
1. Gravitational Force (mg): Acts vertically downward.
2. Normal Reaction Force (N): Acts perpendicular to the surface of the banked road.
3. Frictional Force (if present,[math]F_f[/math]): Acts parallel to the surface of the road, either assisting or opposing motion.
⇒ Without Friction (Ideal Banking Condition)
For an ideal banked curve (one where no friction is required), the horizontal component of the normal force provides the necessary centripetal force.
Resolving the forces:
Vertical direction (balance with gravity):
[math]Ncosθ = mg[/math]
Horizontal direction (providing centripetal force):
[math]N \sin \theta = \frac{mv^2}{r}[/math]
Dividing both equations:
[math]\tan \theta = \frac{v^2}{rg}[/math]
Thus, the optimal banking angle θ for a given speed v and curve radius r is:
[math]\theta = \tan^{-1} \left( \frac{v^2}{rg} \right)[/math]
At this angle, the curve is designed for a specific speed, meaning a vehicle can take the turn without relying on friction.
⇒ With Friction (Real-World Conditions)
If friction is present, it can either assist in providing centripetal force (for higher speeds) or prevent skidding (for lower speeds). The equation then modifies to:
[math]F_f + N \sin \theta = \frac{mv^2}{r}[/math]
This equation allows vehicles to take turns at speeds lower or higher than the ideal speed without slipping.
⇒ Significance of Banking in Roads and Tracks
– Reduces dependence on friction: Essential in conditions like rain or ice.
– Allows higher speeds: Banking helps vehicles take turns at higher speeds without skidding.
– Enhances safety: Prevents accidents caused by insufficient friction on flat roads.
⇒ Examples of Banked Curves in Real Life
– Highways and expressway ramps
– Race tracks for high-speed vehicles
– Railway tracks on curves
– Velodromes (cycling tracks)
Figure 18 Real time examples of banding
q. Relationship Between Angular and Linear Velocity
Angular velocity (ω) and linear speed (v) are related:
[math]v = \frac{2\pi r}{T} \\
v = \omega r[/math]
Where:
– v = Linear speed (m/s)
– r = Radius (m)
– T = Time period (s)
– ω = Angular velocity (rad/s)
⇒ Interpretation
Larger radius → Greater linear speed at the same angular velocity.
Greater angular velocity → Higher linear speed.