Sliding Friction of Rolling Cylinder · IZhO 2022 #1

Physics · mechanics

Problem

To measure the coefficient of sliding friction, the rolling of a solid homogeneous cylinder on an inclined plane is studied at various angles of inclination of the plane to the horizon. We assume that the force of sliding friction is described by the well-known Coulomb-Amonton law: Fs=μsN(1)F_s = \mu_s N \quad (1) where NN stands for the normal reaction force and μs\mu_s denotes the coefficient of sliding friction. Consider the experimental setup to be a wide flat steel plate, whose angle of inclination to the horizon can be arbitrarily varied. The error in setting the angle of inclination of the plane to the horizon is Δα=0.2°\Delta \alpha = 0.2°. The solid homogeneous steel cylinder can roll down the plate with its axis remaining horizontal all the time. To measure the acceleration, three optical sensors (0, 1, 2) are fixed on the plate. The cylinder moves down between these stands and blocks the light beam so that at the moment of light interruption, an electrical impulse is generated that controls an electronic stopwatch. The following time intervals are measured: - t1t_1 – cylinder motion time interval from sensor 0 to sensor 1 - t2t_2 – cylinder motion time interval from sensor 0 to sensor 2 The instrumental error in measuring time intervals t1t_1 and t2t_2 is equal Δt=2×10−4s\Delta t = 2 \times 10^{-4} s. The distances between the sensors are: - from sensor 0 to sensor 1: S1=(50.0±0.2)cmS_1 = (50.0 \pm 0.2) cm - from sensor 0 to sensor 2: S2=(100.0±0.2)cmS_2 = (100.0 \pm 0.2) cm When making numerical calculations, assume that the free fall acceleration is equal g=9.81m/s2g = 9.81 m/s^2. Table 1. Time intervals of the cylinder motion.
α°t1t_1, st2t_2, s
200.45460.7187
250.39360.6290
300.34620.5589
350.32290.5211
400.33580.5283
450.30840.4911
500.26820.4347
550.28160.4432
600.26000.4113
650.24610.3908
700.23080.3675
750.22180.3542
Theoretical part 1.1 Derive formulas for the acceleration of the cylinder axis in two cases: A) the motion of the cylinder occurs without slipping – acceleration a1a_1; B) when moving along the plate, the cylinder slips – acceleration a2a_2. Express your answers in terms of α\alpha, gg, μs\mu_s. 1.2 Express the maximum angle αcr\alpha_{cr} of inclination of the plate, at which the motion of the cylinder still occurs without slipping, in terms of the coefficient of friction μs\mu_s. Processing of measurement data 1.3 Using the measurement results given in Table 1, calculate the acceleration with which the cylinder axis moved for each angle of inclination of the plate. Put down the formula for the acceleration, according to which the calculations are carried out. Draw a graph of the acceleration versus the angle and with its aid find an approximate value of the critical αcr\alpha_{cr}. 1.4 Carry out the linearization of the obtained dependence, i.e. find such values X(α,a)X(\alpha, a) and Y(α,a)Y(\alpha, a) so that the dependence Y(X)Y(X) turns linear for the both cases when the cylinder moves without slipping and with slipping. Draw a linearized dependence Y(X)Y(X) for all experimental points. 1.5 Using the linearized relationship Y(X)Y(X), calculate the coefficient of friction μs\mu_s between the cylinder and the plate. Estimate the error Δμs\Delta \mu_s of the obtained value. Put down the formulas used in your calculations. 1.6 Using the linearized relationship Y(X)Y(X), calculate the value of the critical angle αcr\alpha_{cr}, at which the cylinder starts to slip. Estimate the error of the obtained value. Put down the formulas used in your calculations.

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