Exercises: Rotational Kinetic Energy and Rolling
A flywheel has a rotational inertia of $0.40$ kg$\cdot$m$^2$ and spins at $30$ rad/s. Calculate its rotational kinetic energy.
A flywheel has a rotational inertia of $0.40$ kg$\cdot$m$^2$ and spins at $30$ rad/s. Calculate its rotational kinetic energy.
A solid sphere of mass $2.0$ kg rolls without slipping on a flat floor with a center-of-mass speed of $3.0$ m/s. Identify the two kinds of kinetic energy the sphere has,…
A solid sphere of mass $2.0$ kg rolls without slipping on a flat floor with a center-of-mass speed of $3.0$ m/s. Identify the two kinds of kinetic energy the sphere has, and calculate its translational kinetic energy.
A wheel of radius $0.25$ m rolls without slipping. Write the relationship between the speed $v$ of its center of mass and its angular velocity $\omega$. Then find…
A wheel of radius $0.25$ m rolls without slipping. Write the relationship between the speed $v$ of its center of mass and its angular velocity $\omega$. Then find $\omega$ when $v = 5.0$ m/s.
A hoop of mass $1.5$ kg and radius $0.40$ m rolls without slipping with a center-of-mass speed of $4.0$ m/s. For a hoop, $I = MR^2$. Calculate (a) the translational…
A hoop of mass $1.5$ kg and radius $0.40$ m rolls without slipping with a center-of-mass speed of $4.0$ m/s. For a hoop, $I = MR^2$. Calculate (a) the translational kinetic energy, (b) the rotational kinetic energy, and (c) the total kinetic energy.
A uniform solid disk of mass $3.0$ kg and radius $0.20$ m, with $I = \tfrac{1}{2}MR^2$, is released from rest and rolls without slipping down a ramp, dropping a vertical…
A uniform solid disk of mass $3.0$ kg and radius $0.20$ m, with $I = \tfrac{1}{2}MR^2$, is released from rest and rolls without slipping down a ramp, dropping a vertical height of $0.60$ m. Use energy conservation with $g = 9.8$ m/s$^2$ to find the speed of its center of mass at the bottom.
A motor spins a flywheel of rotational inertia $0.80$ kg$\cdot$m$^2$ from rest to $50$ rad/s. (a) Find the work the motor does on the flywheel, assuming no friction…
A motor spins a flywheel of rotational inertia $0.80$ kg$\cdot$m$^2$ from rest to $50$ rad/s. (a) Find the work the motor does on the flywheel, assuming no friction losses. (b) If this took $4.0$ s, find the average power delivered.
Jayden releases a solid disk ($I = \tfrac{1}{2}MR^2$) and a thin hoop ($I = MR^2$) from rest at the top of the same ramp, a vertical height of $h = 0.80$ m above the…
Jayden releases a solid disk ($I = \tfrac{1}{2}MR^2$) and a thin hoop ($I = MR^2$) from rest at the top of the same ramp, a vertical height of $h = 0.80$ m above the bottom. Both have the same mass and radius and roll without slipping. (a) Derive an expression for each object's center-of-mass speed at the bottom in terms of $g$ and $h$, and calculate both values with $g = 9.8$ m/s$^2$. (b) Determine which object reaches the bottom first and explain why, in terms of how the energy is divided between translation and rotation. (c) State whether the result depends on the mass or radius of the objects.
A uniform solid sphere ($I = \tfrac{2}{5}MR^2$) of mass $0.50$ kg and radius $0.05$ m rolls without slipping along a level track at $2.0$ m/s and then rolls up a ramp.…
A uniform solid sphere ($I = \tfrac{2}{5}MR^2$) of mass $0.50$ kg and radius $0.05$ m rolls without slipping along a level track at $2.0$ m/s and then rolls up a ramp. Assume no energy is lost. (a) Find the sphere's total kinetic energy on the level track and the fraction that is rotational. (b) Find the maximum vertical height the sphere reaches on the ramp, using $g = 9.8$ m/s$^2$. (c) Compare this height with the height a block sliding on a frictionless ramp with the same initial speed would reach, and explain the difference.