A hoop is rolling without slipping along a horizontal surface with a forward speed of 5.50 m/s when it starts up a ramp that makes an angle of 25.0° with the horizontal.What is the speed of the hoop after it has rolled 3.00 m up as measured along the surface of the ramp?
A)4.22 m/s
B)1.91 m/s
C)2.06 m/s
D)3.79 m/s
E)8.02 m/s

Question

Quizplus · Accepted Answer

The hoop is rolling without slipping, which means all its kinetic energy is conserved as it moves up the ramp, transforming from kinetic to potential energy. The speed of the hoop decreases as it climbs due to the conversion of some of its kinetic energy into gravitational potential energy. The energy conservation principle can be applied here:$$ \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}mv_f^2 + \frac{1}{2}I\omega_f^2 + mgh $$For a hoop, $ I = mr^2 $ and $ \omega = \frac{v}{r} $, where $ I $ is the moment of inertia, $ \omega $ is the angular velocity, $ v $ is the linear velocity, and $ r $ is the radius of the hoop. Since the hoop rolls without slipping, $ v = r\omega $, and the equation simplifies because the terms involving $ I $ and $ \omega $ on both sides cancel out due to the relationship between $ v $ and $ \omega $. Thus, we are left with conservation of mechanical energy in the form of linear kinetic energy and gravitational potential energy:$$ \frac{1}{2}mv^2 = \frac{1}{2}mv_f^2 + mgh $$Solving for $ v_f $ (the final velocity), we need to find $ h $, the height gained on the ramp. This can be found using trigonometry since $ h = s \sin(\theta) $, where $ s = 3.00 $ m is the distance along the ramp and $ \theta = 25.0° $ is the angle of the ramp. Plugging in the values:$$ h = 3.00 \sin(25.0°) $$$$ h \approx 1.27 \, \text{m} $$Using the conservation of energy equation and solving for $ v_f $:$$ \frac{1}{2}(5.50)^2 = \frac{1}{2}v_f^2 + 9.81 \times 1.27 $$$$ v_f^2 = (5.50)^2 - 2 \times 9.81 \times 1.27 $$$$ v_f \approx 4.22 \, \text{m/s} $$Therefore, the speed of the hoop after it has rolled 3.00 m up the ramp is approximately 4.22 m/s.

A Hoop Is Rolling Without Slipping Along a Horizontal Surface