Physics
Work, Energy, and Power
Work done by Spring Force
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⚡ Quick Summary
The work done by a spring force depends on the initial and final compression or elongation of the spring. The work can be positive or negative depending on the direction of the displacement relative to the force.
W = -1/2 kx^2 (when spring goes from natural length to compressed/elongated state)
W = 1/2 kx^2 (when spring goes from compressed/elongated state to natural length)
When a spring is compressed or elongated from its natural length, it exerts a force. The work done by this spring force is calculated based on the initial and final positions of the spring. If the spring is initially at its natural length (x=0) and is then compressed or elongated by an amount 'x', the work done by the spring force is -1/2 kx^2. Conversely, if the spring returns to its natural length from a compressed or elongated state, the work done by the spring force is 1/2 kx^2. 'k' represents the spring constant. The sign of the work depends on whether the spring force and displacement are in the same or opposite directions.
Work Done When Force is Perpendicular to Velocity
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If a force is always perpendicular to the velocity of an object, the work done by that force is zero.
W = 0 (if F is perpendicular to v)
If a force F is perpendicular to the velocity v of a particle at all times, then the work done by the force is zero. This is because the dot product of force and displacement (F . dr) is zero, as dr is parallel to v. A common example is circular motion, where the tension in the string (force) is always perpendicular to the velocity of the particle, resulting in zero work done by the tension.
Work Done by a Constant Force
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When a constant force acts on an object, the work done is the product of the force's magnitude, the displacement's magnitude, and the cosine of the angle between them. The work done only depends on the initial and final positions.
W = Fd cos(theta)
W = mgh (work done by gravity)
When a force is constant in both magnitude and direction, the work done by the force is W = Fd cos(theta), where F and d are the magnitudes of the force and displacement, respectively, and theta is the angle between them. The work done depends only on the initial and final positions and not on the path taken. The work done in a round trip is zero. Gravity is an example of such force. The work done by gravity on a particle of mass m is mgh, where h is the vertical height descended by the particle.
Work-Energy Theorem for a System of Particles
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The total work done on a system of particles equals the change in the system's kinetic energy.
N/A
The work-energy theorem is extended to systems of particles where the total work done on all the particles is equal to the change in the total kinetic energy of the system. To change the kinetic energy of a particle, we have to apply a force on it and the force must do work on it.
Work Done by a Force
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Work is done when a force causes a displacement. The amount of work depends on the force and the distance over which it acts.
W = Fd cos(theta)
- Work is done by a force on an object if the point of application of the force moves on the object.
- No work is done by a force if the force is always perpendicular to its velocity or acceleration, or if the object is stationary but the point of application of the force moves on the object, or if the object moves in such a way that the point of application of the force remains fixed.
- Work done by spring force in a displacement is (1/2)kx^2, where k is the spring constant and x is the displacement.
Kinetic Energy
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Kinetic energy is the energy of motion.
KE = (1/2)mv^2
- Kinetic energy of a particle continuously increases with time if the resultant force on the particle is at an angle less than 90 degrees all the time or if the magnitude of its linear momentum is increasing continuously.
Conservation of Energy
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The total mechanical energy of a system remains constant if only conservative forces are acting.
KE_initial + PE_initial = KE_final + PE_final
- The total mechanical energy of a particle is conserved only if the forces acting on it are conservative.
Work Done
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Work is done when a force causes displacement. The amount of work depends on the force, displacement, and the angle between them.
W = Fd cos θ; W = ∫ **F** ⋅ d**r**
Work done by a force is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement. If a force **F** acts on an object causing a displacement **d**, the work done *W* by the force is given by: W = **F** ⋅ **d** = Fd cos θ, where θ is the angle between the force and the displacement vectors.
Work is a scalar quantity. The SI unit of work is the joule (J), where 1 J = 1 N⋅m.
If the force is variable, the work done is calculated by integrating the force over the displacement: W = ∫ **F** ⋅ d**r**
Potential Energy
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Potential energy is stored energy that can be converted into kinetic energy. Gravitational potential energy depends on height, and elastic potential energy depends on the spring's compression or extension.
U_gravitational = mgh; U_elastic = (1/2)kx²
Gravitational Potential Energy: The gravitational potential energy (U) of an object of mass 'm' at a height 'h' above a reference point is given by U = mgh, where 'g' is the acceleration due to gravity.
Change in Potential Energy: ΔU = mg(h₂ - h₁), where h₂ and h₁ are the final and initial heights, respectively.
Elastic Potential Energy: The elastic potential energy stored in a spring with spring constant 'k' when it is stretched or compressed by a distance 'x' from its equilibrium position is given by U = (1/2)kx².
Kinetic Energy
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Kinetic energy is the energy of motion. It depends on the mass and speed of the object.
K = (1/2)mv²
Kinetic Energy (K): The kinetic energy of an object of mass 'm' moving with a speed 'v' is given by K = (1/2)mv².
Work-Energy Theorem
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The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
W_net = ΔK = (1/2)mv_f² - (1/2)mv_i²
Work-Energy Theorem: The work done by all forces acting on a particle is equal to the change in the particle's kinetic energy. W_net = ΔK = K_final - K_initial = (1/2)mv_f² - (1/2)mv_i², where v_f is the final speed and v_i is the initial speed.
Power
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Power is the rate at which work is done or energy is transferred.
P = W/t; P = **F** ⋅ **v**; 1 hp = 746 W
Power (P): Power is defined as the rate at which work is done or energy is transferred. P = W/t, where W is the work done and t is the time taken. Also, P = **F** ⋅ **v**, where **F** is the force and **v** is the velocity.
The SI unit of power is the watt (W), where 1 W = 1 J/s.
Horsepower (hp): 1 hp = 746 W
Circular Motion and Related Concepts
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This section provides solutions to problems involving concepts like tension in strings during circular motion, spring-mass systems, and projectile motion in the context of energy conservation.
['Tension in string at lowest point: T = mg + mv^2/R', 'Condition for completing vertical circle: v >= sqrt(5gR)', 'Potential energy stored in a spring: (1/2)kx^2', 'Conservation of energy: KE_initial + PE_initial = KE_final + PE_final', 'Centripetal force: mv^2/R']
This section includes problems and solutions related to: 1. **Tension in a string during circular motion:** Calculating tension at different points in a vertical circle. 2. **Spring-mass systems:** Determining compression/extension, velocity, and equilibrium positions. 3. **Projectile motion and energy conservation:** Combining projectile concepts with energy principles. 4. **Angle of deviation:** Calculating angle of deviation for objects on circular paths. 5. **Normal reaction:** Finding normal force acting on objects under various conditions.
Conservation of Linear Momentum and Energy in Spring-Mass Systems
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When dealing with systems involving springs and masses, the principles of conservation of linear momentum and energy are crucial. By analyzing the system's initial and final states, we can determine velocities, extensions, and other relevant parameters. In the absence of external forces (like friction), the total mechanical energy of the system remains constant.
['Conservation of Linear Momentum: m1v1 + m2v2 = (m1 + m2)V', 'Kinetic Energy: KE = 1/2 * mv^2', 'Spring Potential Energy: PE = 1/2 * kx^2', 'Pseudo Force: F = ma (in an accelerated frame of reference)']
- Conservation of Linear Momentum: In a closed system (no external forces), the total linear momentum remains constant. This principle is particularly useful when analyzing collisions or systems where internal forces act, such as spring-mass systems. Mathematically, this is represented as: Initial momentum = Final momentum
- Conservation of Energy: In the absence of non-conservative forces (like friction), the total mechanical energy (kinetic + potential) of a system remains constant. For a spring-mass system, the potential energy is stored in the spring when it's compressed or stretched. Mathematically: Initial Energy = Final Energy (Kinetic Energy + Potential Energy)
- Center of Mass Frame: Analyzing the motion of a system from the center of mass frame can simplify the problem, especially when dealing with internal forces. In an accelerated center of mass frame, pseudo forces need to be considered.
Potential Energy
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Energy stored by an object due to its position or configuration.
U = mgh (gravitational), U = (1/2)kx^2 (spring)
Discusses potential energy in relation to changes in rigid body motion, elastic potential energy, and potential energy of a spring.