# Rotational Motion - I have the answer but I don't know how to get there

• jbcaviness
In summary, the problem involves a bike wheel with a torque of 1 N.m applied to it. The wheel has a radius of 35 cm and a mass of 0.75 kg. Treating the wheel as a hoop, the angular acceleration is 10.9 rad/s^2. If the wheel starts from rest, its angular speed after 5 seconds is 54.4 rad/s. In those 5 seconds, the wheel does 21.7 turns. The relationship between torque, moment of inertia, and angular acceleration is that torque is directly proportional to the product of moment of inertia and angular acceleration.
jbcaviness
A torque of 1 N.m is applied to a bike wheel of radius 35 cm and mass 0.75 kg. Treating the wheel as a hoop (I=MR^2)

B. If the wheel starts from rest, what is its angular speed after 5 seconds? Answer - 54.4 rad/s
C. How many turns does it do in those 5 seconds? Answer 21.7 turns

I'm trying to study for my exam, and I'm stuck on this problem for our practice test. My professor gave us answers, but I'm just at a loss as to how to get the answers. Can anyone explain how to solve this problem? Thanks!

What is the relationship between torque, moment of inertia and angular acceleration?

To solve this problem, you need to use the equation for rotational motion: torque = moment of inertia x angular acceleration (τ = Iα). In this case, the moment of inertia (I) for a hoop is equal to the mass (M) multiplied by the radius squared (R^2). So, we can rewrite the equation as τ = MR^2α.

Now, we know that the torque (τ) is 1 N.m and the mass (M) is 0.75 kg. The radius (R) is given as 35 cm, but we need to convert it to meters by dividing by 100. So, R = 0.35 m. Plugging these values into the equation, we get:

1 N.m = (0.75 kg)(0.35 m)^2α

Solving for α, we get α = 10.9 rad/s^2. This is the answer to part A.

For part B, we can use the equation ω = ω0 + αt, where ω0 is the initial angular speed (which is 0 since the wheel starts from rest), α is the angular acceleration we just calculated, and t is the time (5 seconds). So, ω = 0 + (10.9 rad/s^2)(5 s) = 54.4 rad/s.

For part C, we can use the equation θ = θ0 + ω0t + 1/2αt^2, where θ is the angular displacement, θ0 is the initial angular displacement (which is 0 since the wheel starts from rest), ω0 is the initial angular speed (also 0), α is the angular acceleration we calculated, and t is the time (5 seconds). Since we are looking for the number of turns, we need to convert the angular displacement from radians to turns by dividing by 2π. So, θ = (0) + (0)(5) + 1/2(10.9 rad/s^2)(5 s)^2 = 21.7 turns.

I hope this explanation helps you understand how to solve this problem. Remember to always use the correct equations and units when solving rotational motion problems. Good luck on your exam!

## 1. What is rotational motion?

Rotational motion is the movement of an object around an axis or center point.

## 2. How is rotational motion different from linear motion?

Rotational motion involves movement around a fixed point, while linear motion involves movement in a straight line.

## 3. What is the difference between angular velocity and angular acceleration?

Angular velocity is the rate at which an object rotates, while angular acceleration is the rate at which an object's angular velocity changes over time.

## 4. What are some real-world examples of rotational motion?

Some examples of rotational motion include the rotation of the Earth on its axis, the spinning of a top, and the movement of a Ferris wheel.

## 5. How is rotational motion measured?

Rotational motion is typically measured in units such as revolutions per minute (RPM) or radians per second (rad/s).

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