How does the weight force Wfg = mg do positive
or negative work depending on the direction an object moves?
Wfg = ± m · g · h
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A Student Asked…
"Why isn't Wfg on the formula sheet?"
a
Formula sheet section (a):Fg = m · g — the weight force is already defined there.
b
Formula sheet section (b):W = F · d — the general work formula is there too.
Substitute F = mg and d = h
→ you get W = mgh instantly.
c
Formula sheet section (c):PEg = m · g · h is literally
Wfg written as stored energy. Going UP stores +mgh of potential energy;
going DOWN releases it (−mgh). Same formula, different name.
!
Conclusion: Wfg = mgh is not missing — it is embedded in
three places on the formula sheet. Physics avoids listing derivable results as
separate entries. Sign tells you direction;
the next slides explain exactly when it's + or −.
The Foundation
Work = Force × Displacement
This is the master formula. Everything else is a special case of it.
W = F · d · cos θ
F
F — the force acting on the object (Newtons)
d
d — the displacement of the object (metres)
θ
θ — angle between force and displacement direction.
For vertical motion with vertical force: θ = 0° (same direction → cos 0° = +1)
or θ = 180° (opposite → cos 180° = −1).
Key insight: Only the component of force parallel to displacement does work.
Horizontal motion contributes zero work to the weight force — only the
vertical height h counts.
Derivation
From W = F·d → W = mgh
1
Start with the work formula
W = F · d
2
Weight force equals mass × gravity
F = m · g
3
Vertical displacement equals height
d = h
4
Substitute F and d
W = m · g · h
5
Apply the sign based on direction
↑ +mgh | ↓ −mgh
↑ W = +mgh
↓ W = −mgh
g ≈ 10 m/s² (or 9.8 m/s² for higher precision) is a constant.
Only m and h change problem to problem.
Object Moving Up
Work Done is Positive
↑
The object moves upward by height h.
F
The supporting/applied force acts in the same direction
as the displacement (both upward).
+
cos 0° = +1, so work is positive.
Energy is stored as gravitational potential energy.
W = +F·d = +mgh
Object Moving Down
Work Done is Negative
↓
The object moves downward by height h.
F
The resisting force acts opposite to the displacement
(force up, motion down).
−
cos 180° = −1, so work is negative.
Energy is released — potential energy decreases.
W = −F·d = −mgh
Solved Example 1
Book Lifted from the Floor
Given:
mass m = 4 kg |
height h = 1.5 m |
g = 10 m/s² |
direction: UP ↑
①Write the appropriate formula for upward motion: W = +mgh
②Substitute values: W = 4 × 10 × 1.5
③W = 40 × 1.5 = 60 J
✓ Wfg = +60 J (Positive — object moved UP)
Interpretation: 60 joules of energy were stored as gravitational potential energy
as the book was lifted 1.5 m above the floor.
Solved Example 2
Ball Dropped from a Shelf
Given:
mass m = 3 kg |
height h = 2 m |
g = 10 m/s² |
direction: DOWN ↓
①Write the formula for downward motion: W = −mgh
②Substitute values: W = −(3 × 10 × 2)
③W = −(30 × 2) = −60 J
✗ Wfg = −60 J (Negative — object moved DOWN)
Interpretation: 60 joules of gravitational potential energy were
released as the ball fell 2 m to the floor. This energy converts to kinetic energy.
Solved Example 3
Box Pushed Up a Ramp
Given:
m = 8 kg |
ramp length = 6 m |
angle θ = 30° |
g = 10 m/s²
①Only vertical height counts. Find h: h = d · sin θ = 6 × sin 30° = 6 × 0.5 = 3 m
②Motion is upward, so use: W = +mgh
③W = 8 × 10 × 3 = 240 J
✓ Wfg = +240 J (h = 3 m, not ramp length 6 m)
Summary
Work Done by Weight — At a Glance
↑ Moving Up
Force and displacement in the same direction.
Work is positive — energy stored.
W = +mgh
m = mass (kg) · g ≈ 10 m/s² · h = vertical height (m)
↓ Moving Down
Force and displacement in opposite directions.
Work is negative — energy released.
W = −mgh
Always use the vertical height h, not slant length.
Wfg = F · d = mg · h ⟵ already on the formula sheet as W = F·d
Remember for the test:
The formula is not missing — apply W = F·d with F = mg and d = h.
Always ask: which direction is the motion? Up → +mgh,
Down → −mgh.