A large 22 cm marble rolls from rest down an incline. Its center has a linear acceleration of 2.5 m/s^2...What is the angle of the incline?
It's angular accel is 22.7 rad/s^2 (if that helps and IF I'm correct!).
In order to find the angle of the incline, we can use the relationship between linear and angular acceleration when a rolling object is involved. Let's break down the steps to solve this problem:
1. Determine the linear acceleration: In this case, the given linear acceleration is 2.5 m/s^2.
2. Find the angular acceleration: The angular acceleration is given as 22.7 rad/s^2. This information can be used in a later step to double-check the solution.
3. Identify the relationship: For a rolling object, the linear acceleration a and the angular acceleration α are related by the equation: a = R * α, where R is the radius of the object.
4. Calculate the radius: To find the radius of the marble, we need to convert the given diameter of 22 cm into meters. The radius (R) is half the diameter, so R = 0.11 m.
5. Plug in the values: Using the equation from step 3, we can substitute the known values to solve for the angle of the incline. Rearrange the equation to solve for the angle (θ): θ = arctan(a / g), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
θ = arctan(2.5 / 9.8)
6. Calculate the angle: Using a calculator, evaluate the arctangent of 2.5 / 9.8, and you'll find that the angle of the incline is approximately 14.5 degrees.
So, the angle of the incline is approximately 14.5 degrees.