Physics Problem: The Grandfather Clock's Time Warp

A grandfather clock keeps time through the steady swinging of its pendulum. Because the brass rod that forms the pendulum expands when heated, a summer heatwave changes its length, its period, and therefore the accuracy of the clock. Below is the full solution.

PART 1: NEW LENGTH OF THE BRASS PENDULUM

The change in length due to thermal expansion is given by:
ΔL = α · L₀ · ΔT

where:
α = 19.0 × 10⁻⁶ (°C)⁻¹ (coefficient of linear expansion for brass)
L₀ = 1.000 m (original length at 20.0°C)
ΔT = 35.0°C − 20.0°C = 15.0°C

ΔL = (19.0 × 10⁻⁶) × (1.000) × (15.0)
ΔL = 285 × 10⁻⁶ m = 2.85 × 10⁻⁴ m

New length:
L = L₀ + ΔL = 1.000 m + 0.000285 m = 1.000285 m

The pendulum lengthens to approximately 1.000285 meters.

PART 2: NEW PERIOD OF THE PENDULUM AT 35.0°C

The period of a simple pendulum is:
T = 2π√(L/g)

First, find the original period at 20.0°C (L₀ = 1.000 m):
T₀ = 2π√(1.000 / 9.81)
T₀ = 2π√(0.101937)
T₀ = 2π × 0.319276
T₀ = 2.006066 s ≈ 2.00607 s

Now the new period at 35.0°C (L = 1.000285 m):
T = 2π√(1.000285 / 9.81)
T = 2π√(0.101966)
T = 2π × 0.319321
T = 2.006352 s ≈ 2.00635 s

The new period is approximately 2.00635 seconds, slightly longer than the original 2.00607 seconds. The increase is about 0.000286 seconds per swing.

A useful shortcut: for small changes, the fractional change in period is half the fractional change in length:
ΔT_period / T₀ ≈ (1/2)(ΔL / L₀) = (1/2)(2.85 × 10⁻⁴) = 1.425 × 10⁻⁴

PART 3: TIME GAINED OR LOST IN 24 HOURS

Because the pendulum is now longer, each oscillation takes more time. A slower pendulum means the clock ticks too slowly, so the clock LOSES time.

Method using fractional change:
The fractional slowing of the clock equals the fractional increase in period:
ΔT_period / T₀ ≈ 1.425 × 10⁻⁴

Time lost in one day:
Δt = (1.425 × 10⁻⁴) × 86,400 s
Δt = 12.3 s

So the clock loses approximately 12.3 seconds every 24 hours.

Verification by counting swings:
Number of true periods that should occur in a day (based on calibrated T₀):
N = 86,400 / T₀ = 86,400 / 2.006066 = 43,069.4 swings

Time the clock actually displays after these swings (using new period T):
Each swing now takes 2.006352 s, but the clock "believes" each swing is T₀ = 2.006066 s.
After the real day of 86,400 s, the number of actual swings is:
N_actual = 86,400 / 2.006352 = 43,063.3 swings
The clock counts these as N_actual × T₀ = 43,063.3 × 2.006066 = 86,387.7 s
Time lost = 86,400 − 86,387.7 ≈ 12.3 s

Both methods agree: the clock LOSES about 12.3 seconds per day.

PART 4: STEP-BY-STEP EXPLANATION OF THE PHYSICAL PRINCIPLES

Step 1 — Thermal expansion of solids.
When a solid is heated, the average distance between its atoms increases because atoms vibrate with greater amplitude in their slightly asymmetric potential wells. For a long thin rod, this manifests mainly as an increase in length, described by linear expansion: ΔL = αL₀ΔT. Brass has α = 19.0 × 10⁻⁶ per °C, so each degree of warming makes the rod about 19 millionths of its length longer.

Step 2 — The heatwave lengthens the pendulum.
Raising the temperature from 20.0°C to 35.0°C (a 15.0°C rise) increases the 1.000 m rod by 0.285 mm to 1.000285 m. Although this is a tiny change, a clock running continuously accumulates the effect over thousands of swings.

Step 3 — Length controls the period.
The period of a simple pendulum, T = 2π√(L/g), depends on the square root of the length. A longer pendulum has a longer period—it swings more slowly. Importantly, the period does NOT depend on the mass or (for small angles) the amplitude; only L and g matter. Since g is essentially constant in a room, only the length change affects timing.

Step 4 — Why the clock loses time.
A grandfather clock advances its hands by a fixed amount for each complete swing, assuming every swing takes exactly the calibrated period (2.00607 s here). When the pendulum lengthens, each real swing takes slightly longer (2.00635 s), so fewer swings occur in a real day than the clock expects. Because the clock counts each slow swing as if it were a normal one, it under-counts elapsed time and therefore runs slow—it loses time.

Step 5 — Quantifying the error.
Using the differential relation ΔT/T = ½(ΔL/L), the fractional timing error equals half the fractional length increase. Multiplying this fraction (1.425 × 10⁻⁴) by the 86,400 seconds in a day gives a loss of about 12.3 seconds per day.

Conclusion:
During the heatwave, the brass pendulum expands to 1.000285 m, its period increases to about 2.00635 s, and the grandfather clock loses approximately 12.3 seconds every 24 hours. This is precisely why high-quality pendulum clocks use temperature-compensating designs (such as mercury-vial or gridiron pendulums made of metals with opposing expansion coefficients) to keep the effective pendulum length—and thus the period—constant despite temperature changes.

1. New pendulum length at 35.0°C:

The temperature increase is:

ΔT = 35.0°C − 20.0°C = 15.0°C

For linear thermal expansion:

L = L₀(1 + αΔT)

Using L₀ = 1.000 m and α = 19.0 × 10⁻⁶ °C⁻¹:

L = 1.000(1 + (19.0 × 10⁻⁶)(15.0))

L = 1.000(1 + 0.000285)

L = 1.000285 m

So the new length of the brass pendulum is 1.000285 m.

2. New period at 35.0°C:

The period of a simple pendulum is:

T = 2π√(L/g)

Using L = 1.000285 m and g = 9.81 m/s²:

T = 2π√(1.000285/9.81)

T ≈ 2.00635 s

For comparison, the original period at 20.0°C was:

T₀ = 2π√(1.000/9.81) ≈ 2.00607 s

So the new period is about 2.00635 s, slightly longer than before.

3. Time gained or lost in 24 hours:

Because the pendulum becomes longer, its period increases. That means each swing takes slightly more time, so the clock runs slow.

The ratio of the old period to the new period is:

T₀/T = √(L₀/L) = √(1.000/1.000285)

T₀/T ≈ 0.9998575

In 86,400 seconds of real time, the clock will indicate:

86,400 × 0.9998575 ≈ 86,387.7 s

Therefore, the clock loses:

86,400 − 86,387.7 ≈ 12.3 s

So the clock will lose about 12.3 seconds in one 24-hour period.

4. Physical explanation:

A brass pendulum expands when its temperature increases. Since brass has a positive coefficient of linear expansion, heating it from 20.0°C to 35.0°C makes the pendulum slightly longer.

The period of a simple pendulum depends on the square root of its length:

T = 2π√(L/g)

This means that a longer pendulum has a longer period. The dependence is not directly proportional, but even a small increase in length causes a small increase in the time for each swing.

A grandfather clock keeps time by counting pendulum swings. It was calibrated at 20.0°C, when the pendulum length was exactly 1.000 m. At 35.0°C, the brass rod expands to 1.000285 m. This makes the period increase from about 2.00607 s to about 2.00635 s.

Since each swing now takes slightly longer, fewer swings occur in a given amount of real time. As a result, the clock runs slow. Over a full day, the small delay per swing accumulates, causing the clock to lose approximately 12.3 seconds.