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Questions and Answers
Consider two vectors, A and B, with a non-zero angle $\theta$ between them. Under what specific condition will the magnitude of their scalar product, $|A \cdot B|$, be maximized relative to the magnitudes of A and B, while still maintaining a non-zero result?
Consider two vectors, A and B, with a non-zero angle $\theta$ between them. Under what specific condition will the magnitude of their scalar product, $|A \cdot B|$, be maximized relative to the magnitudes of A and B, while still maintaining a non-zero result?
- When $\theta$ approaches 0 radians. (correct)
- When $\theta = \pi/4$ radians.
- When the components of $A$ and $B$ are equal in magnitude but opposite in sign.
- When $A$ and $B$ are orthogonal.
The distributive property of the scalar product, expressed as $A \cdot (B + C) = A \cdot B + A \cdot C$, holds true only when vectors B and C are orthogonal
The distributive property of the scalar product, expressed as $A \cdot (B + C) = A \cdot B + A \cdot C$, holds true only when vectors B and C are orthogonal
False (B)
Given two non-zero vectors $A$ and $B$, what specific geometric condition must be satisfied for their scalar product $A \cdot B$ to equal zero?
Given two non-zero vectors $A$ and $B$, what specific geometric condition must be satisfied for their scalar product $A \cdot B$ to equal zero?
A and B must be orthogonal
The work-energy theorem mathematically relates the net work done on an object to its change in ______.
The work-energy theorem mathematically relates the net work done on an object to its change in ______.
Match the following quantities related to work and energy with their corresponding dimensional formulas:
Match the following quantities related to work and energy with their corresponding dimensional formulas:
Considering a scenario where a force $F$ acts on an object causing a displacement $d$, under what specific condition is NO work done by the force on the object?
Considering a scenario where a force $F$ acts on an object causing a displacement $d$, under what specific condition is NO work done by the force on the object?
In a scenario where a weightlifter steadily holds a barbell weighing 200 kg above his head for 5 minutes, the work done on the barbell is non-zero due to the continuous exertion of force by the weightlifter.
In a scenario where a weightlifter steadily holds a barbell weighing 200 kg above his head for 5 minutes, the work done on the barbell is non-zero due to the continuous exertion of force by the weightlifter.
State the work-energy theorem in its most concise form for a single particle system.
State the work-energy theorem in its most concise form for a single particle system.
The SI unit of work and energy is the ______, named after the physicist James Prescott.
The SI unit of work and energy is the ______, named after the physicist James Prescott.
Match each unit of work/energy with its equivalent in Joules:
Match each unit of work/energy with its equivalent in Joules:
A cyclist applies brakes resulting in a force of 300 N provided by the road, directly opposing the motion, leading to a skidding stop over 15 m. What is the magnitude of the work done by the road on the cyclist?
A cyclist applies brakes resulting in a force of 300 N provided by the road, directly opposing the motion, leading to a skidding stop over 15 m. What is the magnitude of the work done by the road on the cyclist?
According to Newton's Third Law, if body A exerts a force on body B, the work done on A by B is always equal in magnitude and opposite in sign to the work done on B by A.
According to Newton's Third Law, if body A exerts a force on body B, the work done on A by B is always equal in magnitude and opposite in sign to the work done on B by A.
Define kinetic energy mathematically in terms of mass, $m$, and velocity, $v$.
Define kinetic energy mathematically in terms of mass, $m$, and velocity, $v$.
The kinetic energy of a system is a measure of the ______ the object can do by virtue of its motion.
The kinetic energy of a system is a measure of the ______ the object can do by virtue of its motion.
Match the object with its typical Kinetic Energy (K):
Match the object with its typical Kinetic Energy (K):
A bullet is fired into a soft plywood, and emerges with 25% of its initial kinetic energy. If the initial speed of the bullet was 300 m/s, what is its emergent speed?
A bullet is fired into a soft plywood, and emerges with 25% of its initial kinetic energy. If the initial speed of the bullet was 300 m/s, what is its emergent speed?
In a scenario where a force varies with position, the total work done can be accurately determined by simple multiplication of the average force by the total displacement.
In a scenario where a force varies with position, the total work done can be accurately determined by simple multiplication of the average force by the total displacement.
Mathematically define work done by a variable force $F(x)$ from an initial position $x_i$ to a final position $x_f$ in one dimension?
Mathematically define work done by a variable force $F(x)$ from an initial position $x_i$ to a final position $x_f$ in one dimension?
In the context of a force-displacement graph, the work done by the force is equivalent to the ______ under the curve.
In the context of a force-displacement graph, the work done by the force is equivalent to the ______ under the curve.
Relate the following work scenarios to the corresponding calculation method:
Relate the following work scenarios to the corresponding calculation method:
A woman initially applies a constant force of 120 N to a trunk over 12 m and then reduces the force linearly with distance to 60 N, moving it an additional 12 m. Calculate the total work done.
A woman initially applies a constant force of 120 N to a trunk over 12 m and then reduces the force linearly with distance to 60 N, moving it an additional 12 m. Calculate the total work done.
The work-energy theorem is a substitute for Newton's Second Law, providing kinematic information without needing force details.
The work-energy theorem is a substitute for Newton's Second Law, providing kinematic information without needing force details.
If the retarding force experienced by a block moving from ( x = 0.2 ) m to ( x = 2.2 ) m is given by ( F = -\frac{k}{x} ), where ( k = 0.8 ) J, provide the expression to find the work done.
If the retarding force experienced by a block moving from ( x = 0.2 ) m to ( x = 2.2 ) m is given by ( F = -\frac{k}{x} ), where ( k = 0.8 ) J, provide the expression to find the work done.
Potential energy quantifies ______ energy by virtue of a body’s position or configuration.
Potential energy quantifies ______ energy by virtue of a body’s position or configuration.
Match the situation with whether it is kinetic or potential energy
Match the situation with whether it is kinetic or potential energy
A ball of mass ( m ) falls from ( h ) m. Just before reaching, its speed is essentially given by the kinematic relation. Express this relationship in terms that show the gravitational potential energy.
A ball of mass ( m ) falls from ( h ) m. Just before reaching, its speed is essentially given by the kinematic relation. Express this relationship in terms that show the gravitational potential energy.
Gravitational potential energy calculations are universal, requiring no local adjustments or datum shifts regardless of context.
Gravitational potential energy calculations are universal, requiring no local adjustments or datum shifts regardless of context.
How does the principle of conservation of mechanical energy relate potential (V) and kinetic (K) energies, mathematically?
How does the principle of conservation of mechanical energy relate potential (V) and kinetic (K) energies, mathematically?
If forces at play conserve, they are mathematically defined via ______.
If forces at play conserve, they are mathematically defined via ______.
Match the object relatedness
Match the object relatedness
If a spring's potential energy is zero, what position does it take
If a spring's potential energy is zero, what position does it take
Flashcards
Energy
Energy
The capacity to do work. Measured in joules in physics.
Scalar product
Scalar product
A mathematical operation on vectors that yields a scalar quantity.
Work done by a force
Work done by a force
The product of the component of force in the direction of displacement and the magnitude of displacement.
Kinetic energy (K)
Kinetic energy (K)
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Work-energy theorem
Work-energy theorem
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Potential energy (V)
Potential energy (V)
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Total mechanical energy
Total mechanical energy
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Potential energy of a spring
Potential energy of a spring
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Power (P)
Power (P)
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Collision
Collision
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Elastic collision
Elastic collision
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Inelastic collision
Inelastic collision
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Study Notes
Introduction
- Work, energy, and power are common terms in everyday language.
- In physics, "work" has a precise definition.
- Energy equates to the capacity to do work.
- Power signifies forceful actions delivered at great speed.
Scalar Product
- The chapter aims to develop an understanding of the three physical quantities: work, energy and power.
- Before understanding the quantities, it is important to understand scalar product of two vectors.
- Physical quantities, like displacement, velocity, acceleration, and force, are vectors.
- There are two ways of multiplying vectors: scalar product and vector product ways.
Scalar Product Defined
- The scalar product (or dot product) of two vectors A and B is denoted as AB.
- AB = A B cos θ, where θ the angle between the vectors.
- Given A, B, and cos θ are scalars, the dot product of A and B is a scalar quantity.
- Vectors A and B have a direction, but their scalar product has no direction.
Geometric Interpretation
- AB = A (B cos θ) = B (A cos θ)
- B cos θ is the projection of B onto A.
- A cos θ is the projection of A onto B.
- AB is the product of the magnitude of A and the component of B along A, or vice versa.
Laws of Scalar Product
- Commutative law: A·B = B·A
- Distributive law: A·(B + C) = A·B + A·C
- Scalar multiplication: A·(λB) = λ(A·B), where λ is a real number.
Scalar Product Using Unit Vectors
- Given A = Axi + Ayj + Azk and B = Bxi + Byj + Bzk, then:
- AB = (Axi + Ayj + Azk)(Bxi + Byj + Bzk) = AxBx + AyBy + AzBz
- A·A = AxAx + AyAy + AzAz = |A|²
- If A and B are perpendicular, A·B = 0.
Notions of Work and Kinetic Energy: The Work-Energy Theorem
- For rectilinear motion under constant acceleration a: v² - u² = 2 as (where u and v are initial and final speeds; s, the distance).
- Multiplying by m/2: ½ mv² - ½ mu² = mas = Fs
- Generalizing to three dimensions: v² - u² = 2 a·d (where a and d are acceleration and displacement vectors).
- Multiplying by m/2: ½ mv² - ½ mu² = m a·d = F·d
Kinetic Energy and Work Defined
- The left side of the equation (½ mv² - ½ mu²) represents the difference in kinetic energy (K).
- Each of the quantities are defined as 'kinetic energy', and is denoted by K.
- The right side (F·d) involves the product of displacement and the component of force along the displacement.
- This quantity is called work, and is denoted by W
- K - K= W
- K and K are the initial and final kinetic energies of the object.
- Work relates to the force and the displacement over which it acts.
- Work IS done by a force on the body over a certain displacement.
Work–Energy (WE) Theorem
- change in kinetic energy of a particle = work done on it by the net force.
Work
- Work relates to the force and the displacement over which it acts.
- Considering a constant force F acting on an object causing a displacement d in the positive x-direction.
- W = (F cos θ)d = F·d
- If there is no displacement, there is no work done, regardless of the force.
- The muscles still contract/relax, using internal energy, which results in fatigue.
Zero Work Examples
- Displacement is zero: a weightlifter holding weight steadily does no work on the load.
- Force is zero: a block moving on a smooth horizontal table experiences no horizontal force and may undergo large displacement.
- Force and displacement are perpendicular: gravitational force on a block moving horizontally does no work.
- Earth's gravitational force does no work on the moon assuming a circular orbit.
Positive and Negative Work
- Work can be positive or negative.
- cos θ is positive if θ is between 0° and 90°; negative if θ is between 90° and 180°.
- Frictional force opposes displacement, with θ = 180°.
- Work done by friction = negative (cos 180° = −1).
Dimensions and Units of Work
- Work and energy share the same dimensions ([ML²T⁻²]).
- The SI unit is the joule (J), named after James Prescott Joule.
- Alternative units used for work and energy: erg (10⁻⁷ J), electron volt (1.6 × 10⁻¹⁹ J), calorie (4.186 J), kilowatt-hour (3.6 × 10⁶ J).
Kinetic Energy
- kinetic energy K of an object = 1/2 mv·v = 1/2 mv² Kinetic energy is a scalar quantity and represents the work an object can do by virtue of its motion.
- Kinetic energy is used to describe the energy in fast flowing streams.
- Kinetic energy lists the energies for various objects is shown in Table 5.2
Work Done by a Variable Force
- The force isn't constant when it's variable on a curve.
- The displacement ∆x, the force F(x) can can be constant ∆W = F(x) ∆x
- The total work done is the some of the successive rectangular areas under the curve can be:
W= ΣF(x) ∆x
- Final positions need small displacements to approach zero, but the sum can increase without any limits as the sum approaches definite value of the area under the curve W = lim Σ ∆x→₀
∫ F(x) dx
Work Done by a Variable Force continued
- "Lim" stands for the limit when ∆x is near zero.
- The definite integral of force over displacement describes work as a variable force.
Work-Energy Theorem for a Variable Force
- The concepts of work and kinetic energy are helpful to prove work energy for a variable force
- Time rate to change the kinetic energy should confine to one dimension dx
- dK = F dx which can be intergrated ∫ dK = ∫Fdx
- K₁ and K, are initial and final kinetic energies corresponding to x₁ and x
- K-K = W
Work Done by Non Conservative Forces
-
a conservative force F has non conservatism and mechanical energy change from the formula as seen below
-
∆E = Fnc∆x, with E as the total mechanical energy where it assume the form of non conserative nature
-
E - E = W
The Concept of Potential Energy
- Potential energy is the 'stored energy' from the position of the body.
- A stretched bow-string possesses potential energy. When it is released, the arrow flies off at a great speed.
- The gravitational force on a ball of mass m is mg Gravitational potential energy of an object = V(h) and is equal to the negative of work done by the gravitational force =- dV(h) = mg dh
Potential Energy Continued
- The negative sign indicates the gravitational force is downward.
- The potential energy is converted to kinetic energy of the object on reaching the ground.
The Conservation of Mechanical Energy
- For simplicity of demonstration, its best to describe conservative forceF
ΔK = F(x) Δx = -ΔV
- The force can be defined such that by the potential energy function V(x).
- The equations apply that ΔK + ΔV= 0.
- Which means that Kinetic energy+Potential energy of a body is a constant
- If there is no kinetic energy, then the equation will be reversed
Collisions
Conservation laws on momentum are typical. In this section, it is useful to apply energy to common events. In the idealized form on two masses can have an affect to the overall collisions. In most situation there is linear dependence on the directions of how the forces act during the collision. If that is the situation.
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