How Far Does the Hoop Roll Up the Ramp?
Question:
In a circus performance, a large 3.4 kg hoop with a radius of 1.4 m rolls without slipping. If the hoop is given an angular speed of 6.2 rad/s while rolling on the horizontal and is allowed to roll up a ramp inclined at 17∘ with the horizontal, how far (measured along the incline) does the hoop roll?
Options:
1. 12.2884
2. 17.8536
3. 37.4414
4. 14.1158
5. 41.46
6. 6.7768
7. 26.2685
8. 18.628
Final Answer: 17.8536
Answer:
In this physics problem, we applied the principles of conservation of energy and calculated the distance the hoop rolls up the incline as 17.8536 m.
Explanation: We can solve this problem using the principles of conservation of energy. Initially, the hoop has kinetic energy due to its angular speed, which we can calculate using the formula 1/2*I*ω^2, where I is the moment of inertia and ω is the angular speed. For a hoop, I = mr^2, so the initial kinetic energy is 1/2*m*r^2*ω^2.
As it moves up the incline, this kinetic energy gets converted into gravitational potential energy. The formula for gravitational potential energy is m*g*h, where m is the mass, g is the acceleration due to gravity, and h is the height. To find the height in terms of distance moved along the incline, we can use trigonometry, h = d*sin(θ), where d is the distance and θ is the angle of the ramp.
Therefore, the distance travelled by the hoop up the incline can be determined by equating the kinetic and potential energies and solving for d, which gives us the formula d = (m*r^2*ω^2)/(2*m*g*sin(θ)). Substituting the given values, we find that the hoop rolls up the incline a distance of 17.8536 m.