Conservation of Energy in Circus Performance

When a large 3.8 kg hoop with a radius of 1.3 m is given an angular speed of 6.7 rad/s and allowed to roll up a ramp inclined at 15 degrees with the horizontal, the conservation of energy principle can be applied to determine the horizontal distance traveled by the hoop.

Total kinetic energy of the ball

The total kinetic energy of the ball is the sum of the translational and rotational kinetic energy of the ball.

K.E(total) = K.E(trans) + K.E(rotational)

K.E(total) = 1/2mv^2 + 1/2Iω^2

K.E(total) = 1/2mv^2 + 1/2(mR^2)ω^2

K.E(total) = 1/2m(ωR)^2 + 1/2(mR^2)ω^2

K.E(total) = mω^2R^2

K.E(total) = 3.8 × (6.7)^2 × (1.3)^2

K.E(total) = 288.28 J

Conservation of energy

The horizontal distance traveled by the hoop is calculated by equating the total kinetic energy of the hoop to the total gravitational potential energy.

K.E = P.E

288.28 = mg(Lsinθ)

288.28 = (3.8 × 9.8) × (L) × sin(15)

288.28 = 9.63L

L = 30 m

Thus, the horizontal distance traveled by the hoop is approximately 30 meters.

By applying the conservation of energy principle, we can accurately calculate the distance traveled by the hoop in the circus performance scenario described above. The principle of conservation of energy is a fundamental concept in physics that allows us to analyze the motion of objects and understand the relationship between kinetic and potential energy.