91Ó°ÊÓ

A student is asked to measure the acceleration of a cart on a "frictionless" inclined plane as in Figure \(5.11,\) using an air track, a stopwatch, and a meter stick. The height of the incline is measured to be \(1.774 \mathrm{cm},\) and the total length of the incline is measured to be \(d=127.1 \mathrm{cm} .\) Hence, the angle of inclination \(\theta\) is determined from the relation \(\sin \theta=1.774 / 127.1 .\) The cart is released from rest at the top of the incline, and its position \(x\) along the incline is measured as a function of time, where \(x=0\) refers to the initial position of the cart. For \(x\) values of \(10.0 \mathrm{cm}, 20.0 \mathrm{cm},\) \(35.0 \mathrm{cm}, 50.0 \mathrm{cm}, 75.0 \mathrm{cm},\) and \(100 \mathrm{cm},\) the measured times at which these positions are reached (averaged over five runs) are \(1.02 \mathrm{s}, 1.53 \mathrm{s}, 2.01 \mathrm{s}, 2.64 \mathrm{s}, 3.30 \mathrm{s},\) and 3.75 \(\mathrm{s}\) respectively. Construct a graph of \(x\) versus \(t^{2},\) and perform a linear least-squares fit to the data. Determine the acceleration of the cart from the slope of this graph, and compare it with the value you would get using \(a^{\prime}=g \sin \theta,\) where \(g=9.80 \mathrm{m} / \mathrm{s}^{2} .\)

An 8.40 -kg object slides down a fixed, frictionless inclined plane. Use a computer to determine and tabulate the normal force exerted on the object and its acceleration for a series of incline angles (measured from the horizontal) ranging from \(0^{\circ}\) to \(90^{\circ}\) in \(5^{\circ}\) increments. Plot a graph of the normal force and the acceleration as functions of the incline angle. In the limiting cases of \(0^{\circ}\) and \(90^{\circ},\) are your results consistent with the known behavior?