Vector review 1 - distinctions
- v is a vector, v is its magnitude, vx and vy are its components
- ONLY v has a direction. v, vx, and vy are scalar (no direction)
- v, v, vx, and vy ALL have the same units.
Pitfalls
- 1 - You can never add a vector to a scalar. Only vector to vector, or scalar to scalar.
- 2 - v is ALWAYS positive.
- 3 - You can divide two scalars. You can NEVER divide two vectors.
Trick
- If v = c w, then v = c×w.
- Example: I experience an 800 N force net force upwards, and that causes me to accelerate 1.2 m/s2 upwards. What is my mass? F = m a → 800 N up = m × 1.2 m/s2 up. Trick: 800 N = m × 1.2 m/s2. Now I can divide by a to get m.
- ↑ Does not work the other direction. Counterexample: 3 N to left does not equal 3 kg × 1 m/s2 to right, even though 3 N = 3 kg × 1 m/s2
Vector review 2 - components
- A vector's direction is always measured counter clockwise, from the +x axis (rightwards)
- vx = v cos(θ), vy = v sin(θ)
- The x component will be positive if the vector has a rightward direction; negative if leftward, 0 if vertical.
- The y component will be positive if the vector has an upward direction; negative if downward, 0 if horizontal.
Example 1: me standing on level ground
- Forces? (normal from floor, gravity from Earth)
- Directions (u,d,l,r)? (normal = up, gravity = down)
- Directions (°)? (normal = 90°, gravity = 270°)
- Normal components? (FNx = 0, FNy = FN = 667 N)
- Gravity components? (Fgx = 0, Fgy = -Fg = -667 N)
- Note - while components can be positive or negative, magnitudes must be positive.
Example 2: me standing on 10° hill
- (rising from left to right)
- Must be one more force. What is it? (static friction)
- Directions (u, d, l, r)? (normal = up+left, gravity = down, friction = up+right)
- Directions (°)? (normal = 95°, gravity = 270°, friction = 5°)
- Let FN = 664 N, Fs = 58 N. Find components of these two forces.
- FNx = 664 N × cos(95°) = -58 N ← Note it must be negative since normal is leftward.
- FNy = 662 N ← Note it must be positive since normal is upward.
- Fsx = 58 N ← Note it must be positive since friction is rightward.
- Fsy = 5 N ← Note it must be positive since friction is upward.
Magnitude and direction
- v = sqrt(vx2 + vy2)
- θ = atan(vy / vx)
- If vx < 0, then add 180° to θ
Find the magnitudes and directions
- ax = 2 m, ay = -1.5 m (a = 2.5 m, θ = 323° or -37°)
- ax = -1.7 m, ay = -4.2 m (a = 4.5 m, θ = 112°)
Vector review 3 - addition
- To add two vectors, you must ALWAYS start by taking components.
- x component of sum = sum of x components
- If a = b + c + ...
- Then ax = bx + cx + ...
Example 1 - together
- Two objects: One at the origin, and two at x=1.3m, y=0.75m.
- Earth pulls one (gravity) with 25 N, and two pulls one with 20 N.
- What is the net force on one?
- A: F2 = 15 N - what are the components?
- What is the direction from 1 towards 2? (1.3 m right, 0.75 m up → atan(0.75/1.3) = 30°)
- F2x = 20 N cos(30°) = 17 N
- F2y = 20 N sin(30°) = 10 N
- Fnetx = Fgx + F2x = 0 + 17 N = 17 N
- Fnety = Fgy + F2y = -25 N + 10 N = -15 N
Example 2 - have them do
- One is at origin, Two is at (-1.5 m, 1.7 m), Three is at (1.5 m, 1.7 m)
- Two pushes One with 14 N. Three pulls One with 7.2 N
- What is the net force?
- A: Direction from One away from Two is 1.5 m right, 1.7 m down → θ2 = atan(-1.7 / 1.5) = 311°
- F2x = 14 N cos(311°) = 9.3 N
- F2y = 14 N sin(311°) = -10.5 N
- Direction from One towards Three is 1.5 m right, 1.7 m up → θ3 = atan(1.7 / 1.5) = 49°
- F3x = 7.2 N cos(49°) = 4.8 N
- F3y = 14 N sin(311°) = 5.4 N
- Fnetx = F2x + F3x = 9.3 N + 4.8 N = 14.1 N
- Fnety = F2y + F3y = -10.5 N + 5.4 N = -5.1 N