Trolley on an incline: what the intercept is telling you
During a first-year mechanics practical a student releases a 0.50 kg trolley from rest on a 1.20 m inclined track and records its acceleration with a motion sensor. The track is levelled, then tilted to five angles, giving:
sin θ: 0.10, 0.20, 0.30, 0.40, 0.50 a (m s⁻²): 0.64, 1.62, 2.60, 3.58, 4.56
A least-squares fit of a against sin θ gives gradient 9.80 m s⁻² and intercept −0.34 m s⁻². Repeated runs at a fixed angle agree to ±0.02 m s⁻². The student writes in the conclusion: 'The gradient equals g, confirming the theory. The intercept is scatter and carries no physical meaning. Since a = g sin θ, acceleration is proportional to sin θ.' The demonstrator asks the student to reconcile the intercept with the repeatability figure before signing the lab book, and points out that the trolley's wheels are not frictionless.
1. Interpret the intercept physically and calculate the coefficient of friction between trolley and track. State any assumption you make and whether it is safe here.
Read the worked answer
Resolve forces along the incline: the driving component is g sin θ and the retarding component is μg cos θ (for wheels, a rolling-resistance term; the algebra is identical), so
a = g sin θ − μg cos θ.
Over 0.10 ≤ sin θ ≤ 0.50, cos θ only falls from 0.995 to 0.866, so within this range the second term is nearly constant. The graph is then a straight line of gradient g and intercept −μg. Hence
μ = 0.34 / 9.80 = 0.035.
The intercept is not scatter. Repeated runs agree to ±0.02 m s⁻², so −0.34 is roughly 17 standard deviations from zero; a purely random effect could not do that. The sign is also systematic: every measured acceleration lies below g sin θ.
Assumption: cos θ ≈ 1 (i.e. friction term taken as constant). The maximum error this introduces is about 13% of the friction term at sin θ = 0.50, i.e. ≈ 0.04 m s⁻² - small compared with the 0.34 m s⁻² intercept itself, so the estimate of μ is safe to about ±0.01.
2. The student claims that acceleration is proportional to sin θ. Test that claim against the data and say what the supportable conclusion is.
Read the worked answer
Proportionality means a = k sin θ: a straight line through the origin, so the ratio a / sin θ must be constant. From the table:
a / sin θ = 6.40, 8.10, 8.67, 8.95, 9.12.
The ratio varies by more than 40% across the range - it is nowhere near constant, and it only approaches 9.8 at the steepest angle. A line through the origin would predict a = 0.98 m s⁻² at sin θ = 0.10, but 0.64 is measured; at sin θ = 0.50 the prediction is 4.90 against 4.56 measured. So the proportionality claim is simply false on these data; a non-zero intercept is visible by eye.
The honest statement is that a is an affine function of sin θ over this range, consistent with a = 9.80 sin θ − 0.34 cos θ. Proportionality is the frictionless limiting case, approached when μg cos θ ≪ g sin θ, i.e. tan θ ≫ μ. With μ = 0.035 that needs θ ≫ 2°; even at sin θ = 0.10 friction removes 35% of the driving force (μ / tan θ = 0.35), which is exactly why the low-angle points ruin the proportionality.
3. Design one experiment that would distinguish an intercept caused by friction from an intercept caused by a zero offset in the motion sensor, and predict the outcome of each.
Read the worked answer
Use the same incline angle and take two measurements. First, give the trolley a push so it runs up the slope and measure the deceleration magnitude a_up; then release it from rest at the same angle and measure a_down. The forces reverse relative to the motion, so
a_up = g sin θ + μg cos θ, a_down = g sin θ − μg cos θ.
The mean of the two is g sin θ and half the difference is μg cos θ.
Prediction if friction is responsible: plotting (a_up + a_down)/2 against sin θ gives a line through the origin, while (a_up − a_down)/2 ≈ 0.34 m s⁻² at every angle - a constant offset in the difference, not in the mean.
Prediction if the sensor zero is offset: every reading is shifted by the same amount, so the offset survives in the mean but cancels exactly in the difference, giving (a_up − a_down)/2 ≈ 0 and a mean line with the same −0.34 m s⁻² intercept.
The discriminator is therefore the difference, not the mean. A second, cruder check is that friction with wheels barely depends on mass, so adding 0.5 kg to the trolley should leave the intercept term roughly unchanged while the gradient term grows - a sensor offset is unaffected by any of this.