A rolling barrel with initial velocity 20 m/s encounters a hill 10m tall to slow it down
1. state where its angular speed is 0
2. state where its rotational kinetic energy will be 0
3. state where its translation kinetic energy will be 0
4. for each of the two flat surfaces compare the translational kinetic energy, rotational kinetic energy, and potential energy. (State which quantities are greater than or less than the other quantities).
To answer these questions, we need to understand the concepts of angular speed, rotational kinetic energy, translational kinetic energy, and potential energy.
1. To find where the angular speed of the rolling barrel is zero, we need to consider the conservation of angular momentum. In the absence of any external torques, the total angular momentum remains constant. As the barrel rolls up the hill, gravity tries to slow it down, resulting in a decrease in angular speed. Therefore, the angular speed of the barrel will be zero when it reaches the highest point of the hill.
2. Rotational kinetic energy depends on both the angular speed and moment of inertia. To find where the rotational kinetic energy is zero, we need to consider the assumptions of an ideal rolling motion. The barrel's rotational kinetic energy will be zero at any point where it comes to a complete stop during the rolling motion. In this case, once the barrel reaches the top of the hill, it will momentarily stop rotating, resulting in zero rotational kinetic energy.
3. Translational kinetic energy depends on the linear velocity of an object. When the rolling barrel reaches the top of the hill, its linear velocity decreases due to gravity slowing it down. At that point, the translational kinetic energy will be zero since it comes to a stop.
4. Let's compare the different energies for each of the two flat surfaces:
- Translational Kinetic Energy (TKE): The object's TKE depends on its linear velocity. As the barrel rolls down the first flat surface (after the hill), its linear velocity will increase, resulting in an increase in TKE. On the second flat surface, where the barrel is no longer descending but continuing at a constant velocity, the TKE will remain constant.
- Rotational Kinetic Energy (RKE): The object's RKE depends on its angular speed and moment of inertia. After the hill, as the barrel rolls down the first flat surface, its angular speed will decrease due to friction and air resistance, resulting in a decrease in RKE. On the second flat surface, where it continues at a constant velocity, the RKE will remain constant.
- Potential Energy (PE): PE depends on the height and mass of the object. After the hill, the barrel's height above the starting point decreases, resulting in a decrease in PE on both flat surfaces.
Therefore, on the first flat surface, the TKE will be greater than zero, the RKE will be less than its initial value, and the PE will be less than its initial value. On the second flat surface, both the TKE and RKE will remain constant, while the PE will continue to decrease.