Lab 5 Practice

Rotational Equilibrium, from the OnRamps Lab 5 worksheet. Turn a meter stick into a beam balance, find the stick's mass, then find the mass of an unknown object. Every run gives new, realistic data.

Goal: use static equilibrium (ΣF = 0 and Στ = 0) to turn a meter stick into a single-beam balance and find the mass of an object. Average several trials and report the standard uncertainty and 95% confidence interval.

Key formulas

τ = rF sinθ = r·mg (string hangs straight down, θ = 90°) Balance: Στ_ccw = Στ_cw Stick's weight acts at its center of mass Part 2: M = m·r_m / r_stick Part 3 (pivot at CM): m_obj = m·r_m / r_obj s = √[Σ(xᵢ − mean)² / (N − 1)] Δx = s / √N 95% CI = mean ± 2Δx

r is always measured from the pivot to where the string (or the center of mass) is. g cancels when you only compare torques, so Parts 2 and 3 can stay in grams and centimeters. Part 1 asks for torques, so use kg, m and g = 9.80 m/s² there.

Rubric check for each part (10 points per skill)

Design: my experiment gives data that actually answer the question. Set-up: my annotated sketch shows the stick, pivot, every mass and its position, and the distances r. Procedure: another group could repeat my steps without asking me anything. Quantities (scored twice): I list every quantity I measure, with units. Repeats: I take enough trials for the same goal to compute s, Δx and a 95% CI.

P1Exploring static equilibrium

Find the stick's center of mass, mount it there, and balance one mass on each side. Then check whether the torques really sum to zero.

Mass 1 hangs left of the pivot (counterclockwise torque), mass 2 hangs right (clockwise). Enter torques in N·m. Boxes turn green within 2%.

#	m₁ (g)	x₁ (cm)	m₂ (g)	x₂ (cm)	τ₁ ccw (N·m)	τ₂ cw (N·m)	Στ = τ₁ − τ₂	% diff

% diff = |τ₁ − τ₂| ÷ average(τ₁, τ₂) × 100. Στ is checked to ±0.0008 N·m and % diff to ±0.3 points, since they're small differences.

Is this evidence in keeping with the principle of static equilibrium?

Show a sample sketch, procedure and measurements

Show the worked solution for this data

P2Find the mass of the meter stick

Move the pivot off the center of mass. Now the stick's own weight, acting at its CM, makes a torque. Balance it with a single hanging mass on the short side.

Each trial uses a new pivot spot and a different hanging mass, which is allowed. r_m = x_p − x_m, r_stick = x_CM − x_p.

Trial	m (g)	Pivot x_p (cm)	Mass at x_m (cm)	r_m (cm)	r_stick (cm)	Stick mass M (g)

Is one trial (one measurement) enough?

Show a sample sketch, procedure and measurements

Show the worked solution for this data

P3Do-it-yourself balance

Clamp the unknown object to one side and balance it with a known mass on the other. Use the same object every trial.

Show a sample sketch, procedure and measurements

Show the worked solution for this data

Q195% confidence intervals

a. What is the 95% confidence interval of the mass in Part 2? b. In Part 3? Use your mass values from the tables above.

mean = Σxᵢ / Ns = √[Σ(xᵢ − mean)² / (N − 1)]Δx = s / √N95% CI = mean ± 2Δx

Quantity	a. Part 2: M (g)	b. Part 3: m_obj (g)

Mean and CI limits are checked to 2%. s and Δx allow 3%, since rounding adds up.

Q2Interpretation of findings

Do your experimental and known values agree for Part 2? For Part 3? Why or why not? The "known" values come from the electronic scale.

1. Is the scale's stick mass inside your Part 2 95% CI?

2. Is the scale's object mass inside your Part 3 95% CI?

3. The beam-balance masses tend to come out a little low. Which error causes that?

Show a sample answer for this data

Show the worked solution for this data

Back to all study guides.