A 310-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 25° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?
rad/s
To find the angular speed of the sphere at the bottom of the slope, we need to use the principles of rotational motion. Here's a step-by-step breakdown of how to approach this problem:
Step 1: Calculate the gravitational potential energy at the top of the slope.
- The gravitational potential energy is given by the formula: PE = mgh, where m is the mass (unknown), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slope. In this case, the height of the slope is given by h = 6.0 m.
- We need to find the mass of the sphere. To do that, we can use the formula for weight: weight = mass * acceleration due to gravity. In this case, the weight of the sphere is given by W = 310 N.
- Rearranging the equation weight = mass * acceleration due to gravity to solve for mass, we have: mass = weight / acceleration due to gravity = 310 N / 9.8 m/s^2.
Step 2: Find the linear velocity at the bottom of the slope.
- The linear velocity can be found using the principle of conservation of energy. At the top of the slope, the sphere has gravitational potential energy. At the bottom of the slope, that potential energy is converted to kinetic energy.
- The gravitational potential energy at the top is equal to the kinetic energy at the bottom. So, we have the equation: PE = KE. Rearranging, we get: mgh = 0.5 * mass * velocity^2.
- Rearranging the equation to solve for velocity, we have: velocity = sqrt(2gh), where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the slope (6.0 m).
Step 3: Convert linear velocity to angular velocity.
- The angular velocity (ω) is related to the linear velocity (v) and the radius of the sphere (r) by the formula: ω = v / r.
- In this case, the radius of the sphere is given by r = 0.20 m, and the linear velocity is the value we calculated in Step 2.
Step 4: Calculate the angular speed.
- The angular speed is equal to the absolute value of the angular velocity. Thus, the angular speed is: angular speed = |angular velocity|.
Now, you can plug in the values and calculate the angular speed using the above steps and formulas.