What linear speed must a 0.0508-kg hula hoop have if its total kinetic energy is to be 0.190 J? Assume the hoop rolls on the ground without slipping.
To achieve a total kinetic energy of 0.190 J, the hula hoop must have a linear speed of approximately 1.54 m/s. Let's dive deeper into the calculation process to understand how this speed is determined.
Understanding Kinetic Energy and Linear Speed
Kinetic Energy Components:
The total kinetic energy (KE) of the hula hoop consists of two components: rotational kinetic energy (KE_rot) and translational kinetic energy (KE_trans). For a rolling hoop without slipping, these two energies are related by the equation:
KE = KE_rot + KE_trans
Calculating Rotational Kinetic Energy:
The rotational kinetic energy of the hoop can be calculated using the formula:
KE_rot = (1/2) Iω^2
where I is the moment of inertia and ω is the angular velocity. For a hula hoop, the moment of inertia is given by:
I = (1/2) MR^2
where M is the mass of the hoop and R is the radius. Since the hoop is rolling without slipping, the linear speed (v) is related to the angular velocity by the equation:
v = ωR
Substituting these expressions into the equation for rotational kinetic energy, we get:
KE_rot = (1/4) Mv^2
Calculating Translational Kinetic Energy:
The translational kinetic energy is given by:
KE_trans = (1/2) Mv^2
Equating Total Kinetic Energy:
Since the total kinetic energy is the sum of rotational and translational kinetic energies, we can write:
0.190 J = (1/4) Mv^2 + (1/2) Mv^2
Combining the terms, we have:
0.190 J = (3/4) Mv^2
Determining Linear Speed:
To find the linear speed (v), we rearrange the equation:
v^2 = (4/3) * 0.190 J / M
Substitute the values, with M = 0.0508 kg, we can calculate:
v^2 = (4/3) * 0.190 J / 0.0508 kg
v^2 = 2.360 J/kg
v = √(2.360 J/kg)
Calculating the square root, we find:
v ≈ 1.54 m/s
Therefore, the hula hoop must have a linear speed of approximately 1.54 m/s to achieve a total kinetic energy of 0.190 J. This calculation demonstrates the interplay between kinetic energy components and linear speed in the motion of a rolling hula hoop.