You live on a planet far from ours. Based on extensive...

You live on a planet far from ours. Based on extensive communication with a physicist on earth, you have determined that all laws of physics on your planet are the same as ours and you have adopted the same units of seconds and meters as on earth. But you suspect that the value of $g,$ the acceleration of an object in free fall near the surface of your planet, is different from what it is on earth. To test this, you take a solid uniform cylinder and let it roll down an incline. The vertical height $h$ of the top of the incline above the lower end of the incline can be varied. You measure the speed $v_{\mathrm{cm}}$ of the center of mass of the cylinder when it reaches the bottom for various values of $h$. You plot $v_{\mathrm{cm}}^{2}$ (in $\mathrm{m}^{2} / \mathrm{s}^{2}$ ) versus $h$ (in $\mathrm{m}$ ) and find that your data lie close to a straight line with a slope of $6.42 \mathrm{~m} / \mathrm{s}^{2}$. What is the value of $g$ on your planet?

Key Concepts

Linear Relationship between Squared Speed and Height

By utilizing energy conservation and the rolling without slipping condition, one can derive a linear equation relating the square of the center of mass speed (v²) to the vertical height (h) from which the object descends. The slope of this linear relationship incorporates the gravitational acceleration and factors arising from the distribution of mass (through the moment of inertia). Understanding this relationship allows one to experimentally determine the value of gravitational acceleration by analyzing the proportionality constant or slope.

Rolling Without Slipping

This constraint links the translational motion of the center of mass with the rotational motion about the center. Specifically, the no-slip condition means that the linear speed (v) of the center of mass is directly proportional to the angular speed (?), via the relation v = ?R. This relationship is critical for combining the expressions for translational and rotational kinetic energies, leading to the derivation of the speed-height relation.

Energy Conservation

This key concept involves the conversion of gravitational potential energy into kinetic energy as an object descends an incline. In the context of rolling motion, the potential energy (mgh) is transformed into a combination of translational kinetic energy and rotational kinetic energy. By applying energy conservation, one can derive relations between the height, speed of the center of mass, and gravitational acceleration.

Rotational Kinetic Energy and Moment of Inertia

For a rolling object, not all kinetic energy contributes to translation; a portion is associated with rotation about its center. The moment of inertia quantifies the resistance of the object to changes in its rotational motion. In the case of a uniform solid cylinder, the moment of inertia (I = 1/2 mR²) is used to determine the rotational kinetic energy, which is essential to correctly partition the energy and relate it to the speed of the center of mass.

Transcript

Please give Ace some feedback