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Physics

Laws of Motion

Forces between Two Surfaces in Contact

Class 11
⚡ Quick Summary
When two things touch, their atoms interact and create forces. Usually, these forces push the objects away from each other. Friction is when the force also acts along the surface of contact. Smooth surfaces have very little friction.
N/A
  • When two bodies are in contact, electromagnetic forces between their atoms result in measurable forces between the bodies.
  • These forces are generally along the common normal (perpendicular) to the surfaces of contact and are repulsive (pushing).
  • Friction is the component of the contact force parallel to the surface of contact.
  • Frictionless surfaces exert forces only perpendicular to the surface. Smooth surfaces are close to frictionless.
  • Contact forces obey Newton's third law: For every action, there is an equal and opposite reaction.

Tension in a String or a Rope

Class 11
⚡ Quick Summary
When you pull on a rope or hang something from it, the rope becomes tight. This tightness is called tension. The rope is actually stretching a tiny bit, and the molecules inside are pulling on each other to hold everything together. That pull is what we call tension.
N/A
  • Tension exists in a string or rope when it is stretched, such as in a tug of war or when supporting a hanging object.
  • Tension arises from the electromagnetic forces between the atoms and molecules within the string.
  • A string or rope under tension exerts electromagnetic forces on the bodies attached at its ends to pull them.

Force due to a Spring

Class 11
⚡ Quick Summary
Springs push or pull depending on whether you squeeze or stretch them. The more you squeeze or stretch, the harder they push or pull. The spring constant tells you how stiff the spring is.
F = k |x - x₀| = kΔx, where:<ul><li>F is the force exerted by the spring.</li><li>k is the spring constant.</li><li>x is the current length of the spring.</li><li>x₀ is the natural length of the spring.</li><li>Δx is the change in length (extension or compression).</li></ul>
  • A spring exerts a force when its length is changed from its natural length.
  • The natural length of a spring is its length when no force is applied.
  • When stretched, a spring pulls on the objects attached to its ends. When compressed, it pushes on the objects attached to its ends.
  • The force exerted by a spring is proportional to the change in its length (extension or compression), provided the change is not too large.

Gravitational Force vs. Electric Force

11
⚡ Quick Summary
Gravity and electricity both exert forces between objects. This section calculates how much stronger the electric force is compared to the gravitational force between two electrons.
Fe/Fg = e^2 / (4 * pi * epsilon_0 * G(me)^2) = 4.17 * 10^42
  • Two neutral objects far away exert negligible gravitational force.
  • When the objects are placed closer they may exert appreciable force.
  • Electric Force: Fe = e^2 / (4 * pi * epsilon_0 * r^2)
  • Gravitational Force: Fg = G(me)^2 / r^2

Action-Reaction Pairs (Newton's Third Law)

11
⚡ Quick Summary
For every force, there's an equal and opposite force. If you push something, it pushes back on you with the same amount of force. These pairs of forces always act on *different* objects.
N/A (Conceptual understanding)
  • Newton's Third Law states that for every action, there is an equal and opposite reaction.
  • Action-reaction pairs involve two forces that are equal in magnitude and opposite in direction, acting on different objects.

Newton's Second Law

Class 11
⚡ Quick Summary
Force equals mass times acceleration! If you push something harder, it accelerates more. If something is heavier, it accelerates less for the same push.
F = ma
  • In symbols : fi a = Fi /m or, Fi = m ai
  • A force F acting on a particle of mass m produces an acceleration F/m in it with respect to an inertial frame.
  • If the force ceases to act at some instant, the acceleration becomes zero at the same instant.
  • Acceleration and Force are measured at the same instant of time.

Inertial Frame

Class 11
⚡ Quick Summary
An inertial frame is a reference frame where Newton's first law (law of inertia) holds true. Basically, it's a frame that's not accelerating.
N/A
  • The inertial frame is already defined by the first law of motion.

Applying Newton's Laws: System Selection

Class 11
⚡ Quick Summary
When using Newton's laws, first clearly define *what* you're looking at (the 'system'). Make sure every part of your chosen system has the same acceleration. You can't mix things that move differently into a single system.
N/A
  • Decide the system on which the laws of motion are to be applied.
  • The system may be a single particle, a block, a combination of two blocks etc.
  • All parts of the system should have identical acceleration.

Applying Newton's Laws: Identifying Forces

Class 11
⚡ Quick Summary
After deciding on your system, list *only* the forces acting *on* that system *from* things that are NOT part of the system. Don't include forces that the system applies to other things!
N/A
  • Make a list of the forces acting on the system due to all the objects other than the system.
  • Any force applied by the system should not be included in the list of the forces.

Newton's Third Law of Motion

Class 11
⚡ Quick Summary
For every action, there's an equal and opposite reaction! If you push something, it pushes back on you with the same force.
F<sub>AB</sub> = -F<sub>BA</sub> (Force exerted by A on B is equal and opposite to the force exerted by B on A)
  • If a body A exerts a force F on another body B, then B exerts a force -F on A.
  • These forces act on *different* bodies. So, when you're drawing a free-body diagram for one object, only *one* of the action-reaction pair appears.
  • Newton’s third law of motion is not strictly correct when interaction between two bodies separated by a large distance is considered.

Tension in a String

Class 11
⚡ Quick Summary
Tension is the pulling force exerted by a string or rope. Imagine a tug-of-war; the tension is how hard each side is pulling on the rope.
T = Mg (Tension in a string supporting a mass M at rest)
  • The tension in a string at a point is the magnitude of the force exerted by one part of the string on the other part at that point.
  • Consider a cross-section of the string. The lower part of the string exerts an electromagnetic force on the upper part, and the upper part exerts an electromagnetic force on the lower part. These two forces have equal magnitude.
  • To find the tension, consider the forces acting on a part of the string or a connected object, and use Newton's laws.

Motion in Non-inertial Frames and Pseudo Forces

Class 11
⚡ Quick Summary
When you're observing motion from a place that's accelerating (like a car that's speeding up), things can seem a bit weird. To make Newton's laws work in these situations, we add an extra 'fake' force called a pseudo force. It's not a real force applied by an object, but it helps us account for the acceleration of our viewpoint.
a<sub>P, S'</sub> = a<sub>P, S</sub> - a<sub>S', S</sub> F = ma + ma<sub>0</sub> (in non-inertial frame)
  • Newton's second law (F=ma) is only directly applicable in inertial (non-accelerating) frames of reference.
  • When observing motion from a non-inertial frame (S'), an additional term needs to be added to Newton's second law to account for the acceleration of the frame itself.
  • This additional term is called a pseudo force (also known as an inertial force).
  • The pseudo force is calculated as -ma0, where 'm' is the mass of the object being observed and 'a0' is the acceleration of the non-inertial frame (S') with respect to an inertial frame (S).
  • When working in a non-inertial frame, the equation becomes: ma = F - ma0, where 'F' is the sum of all the real forces acting on the object.
  • If you analyze the motion from an inertial frame, no pseudo force is needed (a0 = 0).

Pendulum in an Accelerating Car

Class 11
⚡ Quick Summary
Imagine a pendulum hanging in a car that's speeding up. The pendulum will swing backward a bit. We can figure out the angle it makes with the vertical by considering the 'fake' force acting on it due to the car's acceleration.
tanθ = a<sub>0</sub>/g
  • When analyzing the motion of a pendulum inside an accelerating car (a non-inertial frame), we need to consider the pseudo force.
  • The forces acting on the bob of the pendulum are: tension (T) in the string, weight (mg) due to gravity, and the pseudo force (ma0) acting horizontally opposite to the direction of the car's acceleration.
  • By resolving the forces into components and applying equilibrium conditions (since the bob is at rest relative to the car), we can find the angle (θ) the string makes with the vertical.

Equilibrium of a Body Suspended by Strings

Class 11
⚡ Quick Summary
When a body is suspended by strings and at rest, the forces acting on it (tension in the strings and gravity) must balance each other. This means the vector sum of all forces is zero.
T1 cos(alpha) = T2 cos(beta) T1 sin(alpha) + T2 sin(beta) = mg
To solve problems involving equilibrium of a body suspended by strings: 1. **Identify the forces:** Draw a free body diagram showing all forces acting on the body (weight due to gravity, tension in each string). 2. **Resolve forces into components:** Break down each force into its horizontal and vertical components. 3. **Apply equilibrium conditions:** Since the body is at rest, the sum of horizontal components must be zero, and the sum of vertical components must be zero. 4. **Solve the equations:** You will have a system of equations that you can solve for the unknown tensions in the strings.

Equilibrium of Bodies Connected by a String over a Pulley on an Incline

Class 11
⚡ Quick Summary
When objects are connected by a string over a pulley, especially with one on an incline, the tension in the string and the forces (gravity, normal force) must balance for the system to be at rest. The angle of the incline plays a key role.
T = m2g T = m1g sin(theta) N = m1g cos(theta) sin(theta) = m2/m1
To analyze equilibrium in this scenario: 1. **Free Body Diagrams:** Draw separate free body diagrams for each mass, showing all forces acting on them (tension, weight, normal force for the mass on the incline). 2. **Coordinate System:** Choose a convenient coordinate system for each mass. For the mass on the incline, it's often best to align the x-axis with the incline. 3. **Tension:** If the pulley is smooth and the string is light, the tension is the same throughout the string. 4. **Equilibrium Conditions:** Apply Newton's first law (sum of forces equals zero) to each mass in each direction. 5. **Solve:** Solve the resulting equations to find unknowns like tension, the angle of the incline, or the normal force.

Force and Deceleration

Class 11
⚡ Quick Summary
When a force acts to slow down a moving object, it causes deceleration. We can use kinematics equations to relate the force, deceleration, distance, and initial/final velocities.
v^2 = u^2 + 2ax F = ma
When an object experiences a uniform force opposing its motion, it decelerates. To find the magnitude of the force: 1. **Identify Knowns:** Note the initial velocity (u), final velocity (v), and the distance over which the deceleration occurs (x). 2. **Kinematics:** Use a suitable kinematics equation (like v^2 = u^2 + 2ax) to find the deceleration (a). 3. **Newton's Second Law:** Apply Newton's second law (F = ma) to find the magnitude of the force. Remember that the force will be negative if it's opposing the motion.

Constraint Relations

11
⚡ Quick Summary
When objects are connected (like with a string), their motion is linked. If one moves, the other must move in a related way. We use this to find relationships between their velocities and accelerations.
Example: If a ring and a block are connected by a string, (velocity of the ring) * cos(θ) = (velocity of the block), where θ is the angle between the string and the direction of motion of the ring.
Constraint relations arise when objects are connected by strings, rods, or other means, limiting their independent motion. The length of the connecting element (e.g., string) often remains constant, leading to geometric relationships between the displacements, velocities, and accelerations of the objects. For a string connecting two objects, the sum of the lengths of the string segments along the direction of motion remains constant.

Pulley Systems

11
⚡ Quick Summary
Pulleys change the direction of force and can also provide mechanical advantage (making it easier to lift things). The acceleration of different masses in a pulley system is related to each other.
If one mass has acceleration 'a' downwards, and the other is also linked to the same pulley system, its acceleration might be some fraction or multiple of 'a' depending on how the string is arranged.
In a pulley system, the acceleration of different blocks connected by the same string are not necessarily the same. We need to carefully analyze the constraints imposed by the string lengths and the pulley arrangements. If two masses are connected by a string over a pulley, their accelerations are related by the geometry of the setup.

Motion of a Block on a Prism

Class 11
⚡ Quick Summary
When a block slides down a prism which itself can move, we need to consider the forces on both the block and the prism, and how their accelerations are related. Using a pseudo force simplifies the analysis from the prism's perspective.
a = a₀cosθ + gsinθ, N + ma₀sinθ = mgcosθ, N₀sinθ = Ma₀, a₀ = (mg sinθ cosθ) / (M + m sin²θ), a = ((M + m) g sinθ) / (M + m sin²θ)
  • Analysis involves considering the motion of both the block and the prism.
  • For the block, analyze forces in the frame of the prism (non-inertial frame), including normal force (N), gravity (mg), and pseudo force (ma₀).
  • For the prism, analyze forces from the lab frame (inertial frame), including its weight (Mg), the normal force from the block (N), and the normal force from the horizontal surface (N').
  • Resolve forces into components parallel and perpendicular to the incline.
  • Apply Newton's Second Law (F=ma) to both the block and the prism in their respective frames of reference.
  • Use the relationship between the accelerations of the block and the prism to solve for the unknown accelerations.

Inertial Frame of Reference

Class 11
⚡ Quick Summary
An inertial frame is a place where things move in a straight line at a constant speed unless a force acts on them. It's a frame that's not accelerating. It is the basis of Newton's Laws.
None explicitly mentioned, but the concept is foundational to F = ma
  • Definition: A frame of reference in which Newton's laws of motion hold true. This means an object at rest stays at rest, and an object in motion continues in motion with the same speed and in the same direction unless acted upon by a force.
  • An inertial frame is a non-accelerating frame. A frame moving with constant velocity relative to another inertial frame is also an inertial frame.

Pseudo Force

Class 11
⚡ Quick Summary
When you're in an accelerating frame (like a car speeding up), things seem to get pushed around even if no real force is acting on them. That 'push' is the pseudo force. Remember to consider Pseudo Force when analyzing situations in non-inertial frames.
F_pseudo = -ma₀
  • Pseudo force is an imaginary force that appears to act on an object in a non-inertial (accelerating) frame of reference.
  • It is introduced to make Newton's laws applicable in non-inertial frames.
  • The direction of the pseudo force is opposite to the direction of the acceleration of the non-inertial frame.
  • Magnitude of pseudo force F_pseudo = ma₀ where m is the mass of the object and a₀ is the acceleration of the non-inertial frame.

Tension in a String

11
⚡ Quick Summary
Tension is the force exerted by a string or rope when it is pulled tight. It acts along the direction of the string and pulls equally on the objects connected to the string.
T (Tension)
  • Tension is a pulling force.
  • Tension is the same throughout a massless string.
  • If the string passes over a smooth pulley, the tension remains the same on both sides.
  • Tension acts along the direction of the string.

Spring Force

11
⚡ Quick Summary
Springs exert a force proportional to how much they are stretched or compressed. This force always tries to restore the spring to its original length.
F = -kx (Hooke's Law), where k is the spring constant and x is the displacement from equilibrium
  • Spring force is a restoring force.
  • It is proportional to the displacement from the equilibrium position.

Buoyancy

11
⚡ Quick Summary
Buoyancy is the upward force exerted by a fluid (like air or water) on an object submerged in it. It's what makes things float!
B = ρVg, where ρ is the density of the fluid, V is the volume of fluid displaced, and g is the acceleration due to gravity
  • Buoyant force is equal to the weight of the fluid displaced by the object (Archimedes' Principle).
  • Acts vertically upwards.

Atwood Machine

11
⚡ Quick Summary
An Atwood machine consists of two masses connected by a string over a pulley. It's used to study motion under constant acceleration.
a = (m2 - m1)g / (m1 + m2) (acceleration of the system), T = 2m1m2g / (m1 + m2) (tension in the string), where m2 > m1
  • Assumptions: Massless string, smooth pulley.
  • Tension is the same throughout the string.
  • Acceleration is the same for both masses (but direction is opposite).

Net Force and Acceleration

11
⚡ Quick Summary
When an object experiences a net force, it accelerates in the direction of the force. The amount of acceleration is directly proportional to the net force and inversely proportional to the mass.
F = ma (Newton's Second Law), where F is the net force, m is the mass, and a is the acceleration
  • Net force is the vector sum of all forces acting on an object.
  • Acceleration is the rate of change of velocity.

Weight (Apparent)

11
⚡ Quick Summary
Your apparent weight can change in an accelerating elevator. If the elevator accelerates upwards, you feel heavier; if it accelerates downwards, you feel lighter.
W_apparent = m(g + a) (elevator accelerating upwards), W_apparent = m(g - a) (elevator accelerating downwards), where m is the mass, g is the acceleration due to gravity, and a is the acceleration of the elevator.
  • The weighing machine measures the normal force acting on the person.
  • When the elevator accelerates upwards: apparent weight > true weight.
  • When the elevator accelerates downwards: apparent weight < true weight.

Constraints in Motion

Class 11
⚡ Quick Summary
When objects are connected (like by a string or pulley), their motion is linked. If one moves, the other has to move in a related way. We use these relationships to figure out how they accelerate.
Example: If two blocks are connected by a string over a pulley, and the string's length doesn't change, then the sum of the changes in position of the blocks (measured along the string) must be zero. This leads to relationships like a1 = -a2 (where a1 and a2 are the accelerations of the blocks).
Constraints arise when objects are connected by strings, rods, or other means. These connections impose relationships between the positions, velocities, and accelerations of the objects. Analyzing these geometric constraints is crucial for solving problems involving connected objects. For example, the length of a string remains constant, which links the displacements of the objects it connects. Similarly, the velocities and accelerations of the connected objects are also related.

Tension in Strings

Class 11
⚡ Quick Summary
Strings pull! The force a string exerts is called tension. We usually assume strings are light, so the tension is the same everywhere in the string (except when the string is accelerating and has mass). Tension always pulls along the direction of the string.
T is the tension force. If the string is massless, T is constant along its length.
Tension is the pulling force exerted by a string, cable, chain, or similar object on another object. In ideal cases, the string is assumed to be massless and inextensible (doesn't stretch). This means the tension is uniform throughout the string. The tension acts along the direction of the string, pulling on the objects connected to its ends.

Motion with Pulleys

Class 11
⚡ Quick Summary
Pulleys change the direction of force and can sometimes give you a mechanical advantage (making it easier to lift heavy things). When dealing with pulleys, remember to consider the constraints on the motion of the connected objects.
If a single string wraps around multiple pulleys, and the pulleys are massless and frictionless, then the tension in the string is the same throughout. Relationships between accelerations of different parts of the system need to be determined based on the geometry of the setup.
Pulleys are simple machines that change the direction of tension in a string. A system of pulleys can also provide a mechanical advantage, allowing you to lift a heavier load with less force. When analyzing pulley systems, it's important to account for the tensions in the strings, the masses of the objects, and the constraints imposed by the fixed length of the string(s). We assume ideal pulleys are massless and frictionless.

Free Body Diagrams

Class 11
⚡ Quick Summary
Draw a picture of EACH object. Then, draw ALL the forces acting ON that object. This is your Free Body Diagram. It's how you see all the forces to use F=ma!
Not really a formula, but the whole point is to apply ∑F = ma to each direction based on the forces shown in the FBD.
A free body diagram (FBD) is a visual representation of all the forces acting on an object. To draw an FBD: 1. Isolate the object of interest. 2. Represent the object as a point or a simple shape. 3. Draw vectors representing all the forces acting *on* the object. Include gravity, normal forces, tension, friction, applied forces, etc. Do *not* include forces the object exerts on other things. Choose a coordinate system. 4. Label each force vector clearly. FBDs are essential for applying Newton's Laws of Motion.

Newton's Second Law

Class 11
⚡ Quick Summary
Force equals mass times acceleration (F=ma). More force means more acceleration. More mass means less acceleration (for the same force).
∑F = ma (where ∑F is the vector sum of all forces acting on the object, m is the mass, and a is the acceleration).
Newton's Second Law of Motion states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This is a vector equation, so it applies separately to each component of the force and acceleration.

Magnitude of Static Friction

Class 11
⚡ Quick Summary
Static friction can only be so strong. It's limited by the normal force and a coefficient that tells you how 'sticky' the two surfaces are. But, it will only use as much force as necessary to prevent slipping, up to that limit.
f s £ m s N
  • If the bodies do not slip over each other, the force of friction is given by f s £ m s N where m s is the coefficient of static friction between the bodies and N is the normal force between them.
  • The direction and magnitude of static friction are such that the condition of no slipping between the bodies is ensured.

Magnitude of Kinetic Friction

Class 11
⚡ Quick Summary
Kinetic friction depends on the normal force (how hard the surfaces are pressed together) and a coefficient that describes how rough the surfaces are. The rougher the surfaces, the greater the coefficient of friction, and the more friction there is.
f = m k N
  • If the bodies slip over each other, the force of friction is given by f = m k N where N is the normal contact force and m k is the coefficient of kinetic friction between the surfaces.

Laws of Friction

Class 11
⚡ Quick Summary
Friction opposes relative motion. Static friction prevents slipping and kinetic friction slows down sliding objects. How much friction there is depends on the materials (coefficient of friction) and how hard they're pressed together (normal force).
f = m k N, f s £ m s N
  • We can summarise the laws of friction between two bodies in contact as follows :(1) If the bodies slip over each other, the force of friction is given by f = m N where N is the normal contact force and m is the coefficient of kinetic friction between the surfaces. (2) The direction of kinetic friction on a body is opposite to the velocity of this body with respect to the body applying the force of friction. (3) If the bodies do not slip over each other, the force of friction is given by f s £ m s N where m is the coefficient of static friction between the bodies and N is the normal force between them. The direction and magnitude of static friction are such that the condition of no slipping between the bodies is ensured. (4) The frictional force f or f does not depend on the area of contact as long as the normal force N is same.

Equilibrium of a Block on an Inclined Plane with Another Block Connected by a String

Class 11
⚡ Quick Summary
When you have blocks connected by a string on an inclined plane, and one block is on the incline while the other hangs, you need to balance the forces (tension, friction, weight) to figure out if the system stays still or moves. The biggest thing to keep in mind is that friction always opposes motion.
T = μmg, T = Mg(sinθ - μcosθ)
  • Forces on Block m: Tension (T), Normal force (N), Friction (f), and Weight (mg). Vertical Equilibrium: N = mg. Friction can be limiting, so f = μN = μmg.
  • Forces on Block M (on the incline): Tension (T), Normal force (N₂), and Weight (Mg). Components of Weight: Mg sinθ parallel to the incline, and Mg cosθ perpendicular to the incline.
  • Equations for Block M: T + μN = Mg sinθ and N₂ = Mg cosθ which gives T = Mg(sinθ - μcosθ).
  • Condition for No Sliding: If tanθ < μ, the system will not slide for any value of M/m. This means the angle of the incline is small enough, or the friction is large enough, that the blocks won't move.

Minimum and Maximum Force to Keep a Block at Rest Relative to Another Block

Class 11
⚡ Quick Summary
Imagine one block on top of another. If you want to keep the top block from sliding, you need to apply a force to the bottom block. There's a minimum force to prevent it from sliding one way, and a maximum force before it slides the other way. Friction between the blocks is key here.
a_min = g(1-μ)/(1+μ), a_max = g(1+μ)/(1-μ), F = (M+2m)a
  • Forces on Block A (the smaller block): Tension (T) to the right, Friction (f) to the left, Weight (mg) downward, and Normal force (N) upward.
  • System A+B+C: Acceleration 'a' of the whole system is F/(M+2m), where F is the applied force, M is the mass of the bigger block, and m is the mass of smaller blocks.
  • Limiting Friction: When the block is just about to slip, the friction force is at its maximum value: f = μN = μmg.
  • Equations for Block A: T - f = ma. If the friction is limiting, T - μmg = ma. Also, T + μma = mg. Solving gives a = g(1-μ)/(1+μ) for minimum force and a = g(1+μ)/(1-μ) for maximum force.

System of Two Blocks on Inclined Planes Connected by a String

Class 11
⚡ Quick Summary
Two blocks are on slopes, connected by a string. To keep them from moving, the forces (gravity and friction) pulling them down the slopes have to balance. Find the range of masses that allows the system to be at rest.
f_static <= μ_s * N, f_kinetic = μ_k * N
  • Static and Kinetic Friction: The coefficients of static (μs) and kinetic (μk) friction are given. Remember static friction is the force that prevents initial motion, while kinetic friction acts when the object is already moving. μs > μk
  • Forces involved: Analyze forces acting on each block - Weight (mg), Tension (T) in the string, and Friction (f). Resolve weight into components parallel and perpendicular to the inclines.
  • Finding the acceleration: a = (μ -1) / (1 + μ) * g and a = (1 + μ) / (1 - μ) * g when a block is gently pushed.

Acceleration on Inclined Plane with Friction

11
⚡ Quick Summary
When an object slides down a slope, friction fights against the motion, reducing the acceleration. The steeper the slope (larger angle q), the greater the acceleration, and the larger the friction coefficient (m), the smaller the acceleration.
a = g (sin q - m cos q )
The acceleration of a body sliding down an inclined plane with friction is given by a = g (sin q - m cos q ), where g is the acceleration due to gravity, q is the angle of inclination, and m is the coefficient of kinetic friction.

Tension in a String in Atwood Machine (with acceleration)

11
⚡ Quick Summary
When masses are connected by a string over a pulley (Atwood machine) and the system is accelerating, the tension in the string is affected by the acceleration 'a' of the system.
2 m m (g - a) / (m + M)
The tension in the string of an Atwood machine with two masses m and M, and system acceleration a, can be calculated. If lifting a mass m with acceleration a, the tension is greater than just supporting its weight.

Tension in a String in Atwood Machine with electric field

11
⚡ Quick Summary
When a charged mass m is in the electric field E the tension in the string is affected by the electric force QE.
2m (mg - QE)
The tension in the string of an Atwood machine with two masses m and M, and system acceleration a, can be calculated. If lifting a mass m with acceleration a, the tension is greater than just supporting its weight.

Force Needed to Keep a Block Stationary on a Moving Cart

11
⚡ Quick Summary
Imagine a block is pressed against the side of a cart. If the cart accelerates, the block won't fall if enough force is applied. This force depends on the mass of the block, the cart's acceleration, and the friction between the block and the cart.
m mg
The minimum force required to keep a block of mass 'm' from falling when pressed against the side of a cart accelerating horizontally is related to the friction coefficient and the acceleration of the cart. It's essentially balancing the force of gravity with the frictional force.

Force and Friction

11
⚡ Quick Summary
Deals with forces, types of forces (like friction), and Newton's laws of motion.
Frictional Force (Kinetic): f_k = μ_k * N; Frictional Force (Static): f_s ≤ μ_s * N
Includes different types of forces like gravitational, electromagnetic, nuclear, etc. Covers friction (static, kinetic) and laws of friction.

Tension

11
⚡ Quick Summary
The pulling force exerted by a string, cable, or similar object.
N/A
Tension is the force transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends.