What linear speed must a 0.0487-kg hula hoop have if its total kinetic energy is to be 0.179 J? Assume the hoop rolls on the ground without slipping.
To find the linear speed of the hula hoop, we need to relate its total kinetic energy to its mass and velocity.
The total kinetic energy of a rolling hoop can be calculated by considering two types of motion: translational and rotational.
The translational kinetic energy is given by the formula:
KE_trans = (1/2) * m * v^2
where:
KE_trans is the translational kinetic energy,
m is the mass of the hula hoop, and
v is the linear velocity of the hula hoop.
The rotational kinetic energy is given by the formula:
KE_rot = (1/2) * I * ω^2
where:
KE_rot is the rotational kinetic energy,
I is the moment of inertia of the hula hoop, and
ω is the angular velocity of the hula hoop.
Since the hoop is rolling without slipping, we know that the linear velocity is related to the angular velocity by the formula:
v = ω * R
where:
R is the radius of the hula hoop.
Combining these equations, we can express the total kinetic energy as:
KE_total = KE_trans + KE_rot
= (1/2) * m * v^2 + (1/2) * I * ω^2
= (1/2) * m * v^2 + (1/2) * I * (v/R)^2
To solve for the linear velocity (v), we need to know the moment of inertia of the hula hoop. The moment of inertia of a solid hoop rotating about its central axis (perpendicular to the plane of the hoop) is given by:
I = (1/2) * m * R^2
Substituting this moment of inertia into the equation for total kinetic energy, we get:
KE_total = (1/2) * m * v^2 + (1/2) * [(1/2) * m * R^2] * (v/R)^2
Now we can substitute the given values into the equation and solve for v.
Given values:
m = 0.0487 kg
KE_total = 0.179 J
Solving the equation, we have:
0.179 J = (1/2) * 0.0487 kg * v^2 + (1/2) * [(1/2) * 0.0487 kg * R^2] * (v/R)^2
Simplifying the equation, we get:
0.179 J = (0.02435 kg) * v^2 + (0.0060875 kg * R^2) * (v/R)^2
Since we know the relationship between v and R (v = ω * R), we can write:
v/R = ω = v / R
Substituting this into the equation, we have:
0.179 J = (0.02435 kg) * v^2 + (0.0060875 kg * R^2) * (v / R)^2
To solve for v, we can solve this equation numerically by using an iterative method or a numerical solver. Alternatively, we can rearrange the equation to isolate v and solve for it algebraically, but the resulting equation is quite complex.
Therefore, to find the linear speed of the hula hoop, it is recommended to use a numerical solver or an iterative method to obtain the value of v.