Unwinding a string around a spool, what is the angular acceleration?

In summary, the problem involves a light string wrapped around a cylindrical spool with a radius of 0.0195 m and mass of 0.380 kg. A 8.40 kg mass is attached to the free end, causing the string to unwind from the spool. The question is asking for the angular acceleration of the spool, given the value of g as 9.81m/s^2. The incorrect answer of 22241.14162 was obtained by assuming the tension in the string equals the weight of the mass, but the correct approach involves considering the forces acting on each object and solving for the unknowns using two equations.
  • #1
mustang
169
0
Problem 5. given:g=9.81m/s^2.
A light string 1.20 m long is wrapped around a solid cylindrical spool with a redius of 0.0195 m and a mass of 0.380 kg. A 8.40 kg mass is then attached to the free end of the string, causing the string to unwind from the spool.
a. What is the angular acceleration of the spool? Answer in rad/s^2.
Note : I worked on the problem and I got 22241.14162 by using the formula (moment of inertia* angular acceleration= mass*g*radius. However my answer was wrong what did I do wrong?
 
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  • #2


Originally posted by mustang

Note : I worked on the problem and I got 22241.14162 by using the formula (moment of inertia* angular acceleration= mass*g*radius. However my answer was wrong what did I do wrong?
You are assuming that the tension in the string (which is what is pulling on the spool) equals the weight of the mass.

To do this problem correctly, treat each object separately. Consider the forces acting on each body, and write down the equations describing each. You'll have two equations and two unknowns. And the right answer.
 
  • #3


Based on the given information, the angular acceleration of the spool can be calculated using the formula: moment of inertia * angular acceleration = mass * g * radius. In this case, the moment of inertia can be calculated as I = 1/2 * m * r^2, where m is the mass of the spool and r is the radius. Substituting the given values, we get I = 1/2 * 0.380 kg * (0.0195 m)^2 = 3.5715e-5 kg*m^2.

Next, we can rearrange the formula to solve for angular acceleration: angular acceleration = (mass * g * radius) / moment of inertia. Substituting the given values, we get angular acceleration = (8.40 kg * 9.81 m/s^2 * 0.0195 m) / 3.5715e-5 kg*m^2 = 459.917 rad/s^2.

Therefore, the angular acceleration of the spool is approximately 459.917 rad/s^2. It is possible that your answer of 22241.14162 was incorrect because you may have used the incorrect value for the moment of inertia or made a calculation error. Double-checking your calculations can help identify any mistakes.
 

1. What is angular acceleration?

Angular acceleration is a measure of how fast an object's angular velocity is changing. It is defined as the rate of change of angular velocity over time and is typically measured in radians per second squared.

2. How does angular acceleration differ from linear acceleration?

Angular acceleration is the measure of how an object's angular velocity changes, while linear acceleration is the measure of how an object's linear velocity changes. Angular acceleration is typically used to describe the rotation of an object, while linear acceleration is used to describe the movement of an object in a straight line.

3. How is angular acceleration calculated?

Angular acceleration can be calculated by dividing the change in angular velocity by the change in time. This is expressed in the formula: α = (ω2 - ω1) / (t2 - t1), where α is the angular acceleration, ω is the angular velocity, and t is the time.

4. What factors can affect the angular acceleration of a string unwinding around a spool?

The angular acceleration of a string unwinding around a spool can be affected by several factors, including the tension of the string, the mass of the spool, the radius of the spool, and any external forces acting on the string or spool.

5. How does the angular acceleration of a string unwinding around a spool relate to its linear acceleration?

The angular acceleration of a string unwinding around a spool can be used to calculate the linear acceleration of the string. This can be done using the formula: a = αr, where a is the linear acceleration, α is the angular acceleration, and r is the radius of the spool.

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