Scalars & Vectors

Aprenda jogando

Responda a estas perguntas para ganhar energia, depois pesque e explore. Sem precisar de conta.

Para educadores: slides de aula, resumos de revisão prontos para usar de Scalars & Vectors (Physics, AQA) — use na sua aula ou rode o tópico como atividade interativa de turma que seus aprendizes jogam como um jogo ao vivo.

Notas de aula Slides Questões de prática

Notas de aula

Distinguish magnitude from direction

A scalar has magnitude without a spatial direction. Examples include mass, time, energy, temperature, speed and distance. A scalar can still have a negative value, as electric charge can.
A vector has magnitude and direction. Examples include displacement, velocity, acceleration, force and momentum. Specify reference axes or a compass direction in a vector answer.
An arrow represents a vector: its length is proportional to magnitude at the chosen scale and its arrowhead shows direction. Moving a free vector parallel to itself does not change that vector.
The zero vector has magnitude zero and no unique direction. Multiplying a nonzero vector by a negative scalar reverses its direction and multiplies its magnitude by the absolute value of that scalar.

Vector arrows and a resultant

Distance, displacement, speed and velocity

Distance is total path length travelled. Displacement is the vector change in position from the initial point to the final point. A return journey can have positive distance and zero displacement.
For a journey 6 m east followed by 4 m west, distance is 10 m and displacement is 2 m east. Adding path lengths is different from adding signed changes in position.
Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time. They can differ over the same journey.
Constant speed does not guarantee constant velocity: changing direction changes velocity, as in motion round a bend.

A journey on an east–west line

Add and subtract vectors

For collinear vectors, choose a positive direction and add signed components. A force 9 N east combined with 4 N west gives 5 N east.
For head-to-tail addition, place each new vector’s tail at the previous head without changing its length or direction. The resultant joins the initial tail to the final head.
For the parallelogram method, give two vectors a shared tail. The diagonal from that shared tail to the opposite corner is the resultant.
Subtraction A − B means A + (−B). Reverse B before constructing the sum. A scale drawing must use one consistent magnitude scale, with its result converted back to physical units.

Head-to-tail force addition

Calculate perpendicular resultants

For perpendicular components A and B, the resultant magnitude is R = √(A² + B²). Do not add magnitudes directly unless the vectors point in the same direction.
The angle from the A direction toward B satisfies tan θ = B/A. State which axis the angle is measured from and use component signs to choose the correct quadrant.
For 3 N east and 4 N north, R = 5 N and θ = tan⁻¹(4/3) ≈ 53.1° north of east. The angle from north is instead about 36.9°.
AQA calculations here focus on two vectors at right angles; scale constructions may use other angles. Components provide a consistent way to reason about direction.

Two perpendicular forces

Resolve a vector into components

Resolving replaces a vector with perpendicular components that have the same combined effect. It does not create two additional forces alongside the original.
If a force F is at angle θ above the positive horizontal, Fx = F cos θ and Fy = F sin θ. Signs depend on the chosen directions.
If θ is measured from the vertical, the vertical component is F cos θ and the horizontal component is F sin θ. Identify the adjacent side before choosing sine or cosine.
For F = 20 N at 30° above horizontal, Fx ≈ 17.3 N and Fy = 10 N. Recombining these perpendicular components gives the original force.

A 20 N force and its components

Resolve weight on an inclined plane

Weight W acts vertically downward. On a plane inclined at θ to the horizontal, its component down the plane is W sin θ and its component perpendicular into the plane is W cos θ.
The normal contact force acts perpendicular to the surface, away from it. It is not generally vertically upward on a slope. Friction acts along the surface and opposes relative motion or its tendency.
If there is no perpendicular acceleration and no other forces with perpendicular components, R = W cos θ. This is a conditional force balance, not a universal rule for every slope problem.
For a 20 N block on a rough 30° slope at rest with no other forces, the perpendicular component is about 17.3 N and the downslope component is 10 N. A 10 N static friction force acts upslope.

Force balance on a rough slope

Use equilibrium and closed force triangles

A particle is in translational equilibrium when the vector sum of the forces on it is zero. In an inertial frame it has zero acceleration and can be at rest or moving with constant velocity.
For two forces to balance, they must be equal in magnitude and opposite in direction. For three coplanar forces acting at a point, a head-to-tail construction closes into a triangle.
Resolve along two perpendicular axes to check ΣFx = 0 and ΣFy = 0. An equilibrant is equal in magnitude and opposite in direction to the existing resultant.
These particle conditions concern forces acting at a point. A rigid body can also require a separate moment balance; do not assume that zero resultant force alone prevents rotation.

Three coplanar forces

Slides

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

CADASTRE-SE — GRÁTIS ENTRAR

Questões de prática

Prévia grátis — 8 de 40 perguntas. Cadastre-se para ver todas.

1.Which quantity is a vector?
Easy
- ADisplacement
- BDistance
- CMass
- DEnergy
2.Which quantity is a scalar?
Easy
- AForce
- BSpeed
- CVelocity
- DAcceleration
3.A quantity can have a negative value. What can be concluded from the sign alone?
Medium
- AIt cannot be a scalar
- BIt must be a force
- CIt is not enough to decide whether the quantity is a vector
- DIt must be a vector
4.A runner completes one circuit and returns to the starting point. Which statement is correct?
Easy
- ADistance and displacement are both zero
- BDisplacement equals the circumference
- CAverage velocity must equal average speed
- DDisplacement is zero although distance travelled is positive
5.A person walks 6 m east and then 4 m west. What is the displacement from the start?
Easy
- A2 m east
- B10 m east
- C2 m west
- D10 m west
6.A person walks 6 m east and then 4 m west. What distance is travelled?
Easy
- A24 m
- B10 m
- C2 m
- D−2 m
7.A car moves round a bend at constant speed. Which statement can still be true?
Medium
- AIts displacement must be zero
- BIts mass changes because velocity is a vector
- CIts velocity changes because its direction changes
- DIts acceleration must be zero
8.A particle travels 12 m in 3 s and returns to its starting point during that interval. What is its average velocity?
Medium
- A4 m s−1
- B12 m s−1
- C36 m s−1
- D0 m s−1

Mais perguntas

9.Two vector arrows use the same scale and point in the same direction. One is twice as long. What does this represent?
Easy
- ATwice the magnitude with unchanged direction
- BTwice the direction angle
- CA scalar instead of a vector
- DEqual magnitudes
10.What is the effect of multiplying a nonzero vector by −2?
Medium
- AIt becomes a scalar
- BIts magnitude doubles and its direction reverses
- CIts magnitude becomes negative without changing direction
- DIts magnitude halves and its direction reverses
11.Forces of 7 N east and 5 N east act at a point. What is their resultant?
Easy
- A12 N west
- B35 N east
- C12 N east
- D2 N east
12.Forces of 9 N east and 4 N west act at a point. What is their resultant?
Easy
- A13 N east
- B5 N west
- C36 N east
- D5 N east
13.Two equal and opposite forces act at a particle. What is the resultant force?
Easy
- AZero
- BTwice either force
- CA force at right angles to both
- DA negative scalar mass
14.Forces of 3 N east and 4 N north act at a point. What is the magnitude of the resultant?
Medium
- A12 N
- B5 N
- C7 N
- D1 N
15.Forces of 6 N east and 8 N north act at a point. What is the magnitude of the resultant?
Medium
- A2 N
- B48 N
- C10 N
- D14 N
16.Forces of 3 N east and 4 N north act at a point. What is the direction of the resultant, measured north of east?
Medium
- AAbout 36.9°
- B90°
- CAbout 126.9°
- DAbout 53.1°
17.What describes the resultant in a head-to-tail vector addition?
Medium
- AAn arrow from the initial tail to the final head
- BAn arrow joining the two original tails after reversing both vectors
- CThe sum of arrow lengths regardless of direction
- DAn arrow from the final head back to the initial tail
18.In a parallelogram construction with two vectors sharing a tail, which diagonal is the resultant?
Medium
- AA diagonal perpendicular to both input vectors
- BThe diagonal starting at the shared tail
- CThe diagonal joining the two arrowheads
- DEither diagonal with either direction
19.How is vector subtraction A − B performed?
Medium
- AReverse A and add B
- BAlways draw A and B at right angles
- CAdd A to a vector equal in magnitude and opposite in direction to B
- DSubtract the magnitudes and ignore their directions
20.At a drawing scale of 1 cm = 4 N, a resultant measures 3.5 cm. What magnitude does it represent?
Medium
- A0.875 N
- B7.5 N
- C35 N
- D14 N
21.A 20 N force acts 30° above the positive horizontal axis. What is its horizontal component?
Medium
- AAbout 17.3 N
- B10.0 N
- C20.0 N
- DAbout 34.6 N
22.A 20 N force acts 30° above the positive horizontal axis. What is its vertical component?
Medium
- A40.0 N
- B10.0 N
- CAbout 17.3 N
- D20.0 N
23.A 10 N force acts 60° above the positive horizontal axis. What is its horizontal component?
Medium
- A10 N
- B20 N
- C5.0 N
- DAbout 8.66 N
24.A 10 N force acts 60° above the positive horizontal axis. What is its vertical component?
Medium
- A5.0 N
- B10.0 N
- CAbout 17.3 N
- DAbout 8.66 N
25.A force acts at angle θ from the vertical. Which expression gives its vertical component?
Medium
- AF cos θ
- BF sin θ
- CF tan θ
- DF / sin θ
26.Perpendicular components of a force are 5 N and 12 N. What is the force magnitude?
Medium
- A60 N
- B13 N
- C17 N
- D7 N
27.A force has components Fx = −3 N and Fy = +4 N. In which direction does it act?
Medium
- ADownward and to the left
- BHorizontally to the right
- CUpward and to the left
- DUpward and to the right
28.An object rests on a plane inclined at 30° to the horizontal. In which direction does its weight act?
Medium
- APerpendicular into the plane
- BParallel down the plane
- CVertically upward
- DVertically downward
29.What direction does a normal contact force act on an inclined plane?
Medium
- APerpendicular to the plane, away from it
- BAlways vertically upward
- CParallel up the plane
- DVertically downward
30.A block has weight 20 N on a plane inclined at 30° to the horizontal. What is the component of weight down the plane?
Medium
- A40 N
- B10 N
- CAbout 17.3 N
- D20 N
31.A block has weight 20 N on a plane inclined at 30° to the horizontal. What is the magnitude of its weight component perpendicular into the plane?
Medium
- A20 N
- BAbout 34.6 N
- CAbout 17.3 N
- D10 N
32.A block has no acceleration perpendicular to a plane. Only its weight and the normal contact force have perpendicular components. What is the normal force magnitude?
Medium
- AW sin θ
- BW in every case
- CZero in every case
- DW cos θ
33.A block of weight 30 N rests on a rough 30° slope with no other applied forces. What friction magnitude is needed for equilibrium?
Medium
- A15 N
- B30 N
- CAbout 26.0 N
- DZero
34.A particle has zero resultant force. Which statement follows in an inertial reference frame?
Medium
- AIts acceleration is proportional to the sum of force magnitudes
- BIts velocity is constant, possibly zero
- CIt must be stationary
- DIt must have zero speed at that instant
35.Three coplanar forces act at a point in equilibrium. What must their head-to-tail vector construction form?
Medium
- AThree arrows in the same direction
- BA circle whose radius equals the largest force
- CA closed triangle
- DAn open triangle
36.For three coplanar forces at a point, what component condition represents equilibrium?
Medium
- AOnly the horizontal sum must be zero
- BOnly the vertical sum must be zero
- CAll three force magnitudes must be equal
- DThe sums of both perpendicular components are zero
37.Two perpendicular forces of 8 N east and 6 N north act at a point. What additional force would bring the point into equilibrium?
Hard
- A10 N, with components 8 N west and 6 N south
- B10 N northeast
- C14 N southwest
- D2 N south
38.A vector has zero magnitude. Which statement about its direction is correct?
Medium
- AIt points in every direction with unit magnitude
- BIt has no unique direction
- CIt always points east
- DIt must point vertically downward
39.Is pressure a scalar even though a pressure force on a surface has a direction?
Medium
- ANo; pressure must point in the direction of fluid motion
- BOnly negative pressures are scalars
- CYes; pressure is scalar, while the force on a specified surface is a vector
- DNo; every quantity that can produce a force is a vector
40.Two perpendicular vectors have magnitudes 9 units and 12 units. Which pair gives their resultant magnitude and angle from the 9-unit vector toward the 12-unit vector?
Hard
- A21 units and 45°
- B15 units and about 36.9°
- C3 units and 90°
- D15 units and about 53.1°

Resumos de revisão

Scalars have magnitude; vectors have magnitude and spatial direction. A sign alone is not enough to distinguish them. Distance adds path lengths, while displacement adds changes in position. Add vectors with one consistent scale or by perpendicular components. Use R = √(A² + B²) for two perpendicular vectors and state the reference axis for any angle. For an angle from the horizontal, Fx = F cos θ and Fy = F sin θ. On a slope, weight stays vertical and the normal force is perpendicular to the surface. Particle equilibrium requires zero resultant, which allows rest or constant velocity.

Scalar: A quantity described by magnitude without a spatial direction.
Vector: A quantity described by magnitude and direction.
Resultant: The single vector equivalent to a vector sum.
Component: One of a set of vectors that combine to give the original vector, often along perpendicular axes.
Equilibrium: For a particle in an inertial frame, zero resultant force and therefore zero acceleration.

Unlock all 40 questions & more

Crie uma conta grátis para ver todas as perguntas, os slides, flashcards e resumos de revisão deste tópico.

CADASTRE-SE — GRÁTIS ENTRAR

Provas anteriores

A prática com provas anteriores deste tópico chega em breve.

Em breve