Question

The motor of a fan turns a small wheel of radius \(r_{\mathrm{m}}=\) \(2.00 \mathrm{~cm} .\) This wheel turns a belt, which is attached to a wheel of radius \(r_{f}=3.00 \mathrm{~cm}\) that is mounted to the axle of the fan blades. Measured from the center of this axle, the tip of the fan blades are at a distance \(r_{\mathrm{b}}=15.0 \mathrm{~cm} .\) When the fan is in operation, the motor spins at an angular speed of \(\omega=1200\). rpm. What is the tangential speed of the tips of the fan blades?

Short answer

Answer: The tangential speed of the tips of the fan blades can be found using the expression \(v_{b} = 5.0\text{ cm} \cdot ω_{m}\). Substitute the value of \(ω_{m}\) from step 1 to find the value of \(v_{b}\).

Step-by-step solution

Convert angular speed of the motor to radians per second

To begin, we first have to convert the given angular speed of the motor from rotations per minute (rpm) to radians per second. To do this, we'll make use of the conversion factor, where 1 rotation equals 2π radians and 1 minute equals 60 seconds. Given, Angular speed of the motor, \(ω = 1200 \text{ rpm}\) To convert this to radians per second, we will use the conversion factor: Angular speed in radians per second (rad/s) \(= 1200\frac{\text{rotations}}{\text{min}} \cdot \frac{2\pi\text{ rad}}{\text{rotation}} \cdot \frac{1\text{ min}}{60\text{ s}}\)

Find the velocity of the belt on small and large wheels

Next, we will find the linear velocities of the belt on the small wheel (\(v_{m}\)) and the large wheel (\(v_{f}\)). Since the belt connects the small and large wheels, the linear velocities at both wheels will be equal. We can find the linear velocity on the small wheel with the equation \(v_{m} = r_{m}ω_{m}\), where \(r_{m}\) is the radius of the small wheel and \(ω_{m}\) is the angular velocity of the motor in radians per second, calculated in Step 1 . \(v_{m}=r_{m}ω_{m} = 2.00\text{ cm} \cdot ω_{m}\) As the linear velocities are the same, therefore \(v_{f} = v_{m}\).

Find the angular speed of the large wheel

Since we know the linear speed \(v_{f}\), we can determine the angular speed of the large wheel (\(ω_{f}\)) using the equation: \(v_{f}=r_{f}ω_{f}\) , where \(r_f\) is the radius of the large wheel. \(\omega_{f} = \frac{v_{f}}{r_{f}} = \frac{v_{m}}{3.00 \text{ cm}}\)

Find the tangential speed of the tips of the fan blades

Now that we have the angular speed of the large wheel, we can determine the tangential speed of the tips of the fan blades (\(v_{b}\)). We know the distance from the center of the axle to the tip of the fan blades is \(r_{b} = 15 \text{ cm}\). We will use the equation \(v_{b} = r_{b} \omega_{f}\) to find the tangential speed of the tips of the fan blades. \(v_{b} = r_{b}\omega_{f} = 15.0\text{ cm} \cdot \frac{v_{m}}{3.00\text{ cm}}\) We can now substitute the expression of \(v_m = 2.00\text{ cm} \cdot ω_{m}\) into the above equation: \(v_{b} = 15.0\text{ cm} \cdot \frac{(2.00\text{ cm})ω_{m}}{3.00\text{ cm}} =5.0\text{ cm} \cdot ω_{m}\) Finally, substitute the value of \(ω_{m}\) from step 1 into this equation to find the value of \(v_{b}\).

Key concepts

Angular Speed Conversion

Understanding how to convert angular speed into different units is crucial for solving problems related to rotational motion. Angular speed typically measures how quickly something rotates, usually an object like a wheel or a planet, expressed in terms like rotations per minute (rpm) or radians per second (rad/s).

To convert from rpm to rad/s, we multiply the given value in rpm by \( \frac{2\pi}{60} \) because there are \( 2\pi \) radians in one rotation and 60 seconds in one minute. This conversion is essential since rad/s is the standard unit used in most physics equations related to angular motion.

By knowing how to perform this conversion accurately, students are able to switch between different units depending on what is required by the problem they are solving or the system they are working within. It should become a familiar step in problems involving rotational kinematics.

Linear Velocity

Linear velocity, often referred to as tangential speed when dealing with rotational motion, describes how fast an object is moving along a path. In the context of rotation, it's the speed at which a point on the circumference of a rotating object moves. It can be calculated from angular speed using the simple relationship \( v = r\omega \) where \( v \) is the linear velocity, \( r \) is the radius of the rotation, and \( \omega \) is the angular speed in rad/s.

This formula is a cornerstone in understanding how different points on a rotating object move at different speeds despite sharing the same angular speed. For example, in a rotational system like a fan, points closer to the axis of rotation move slower in terms of linear velocity compared to points further out on the blades. This concept is fundamental in many applications, from simple mechanical systems to complex astrophysical phenomena.

Angular Velocity

Angular velocity, symbolized by \( \omega \), measures the rate of change of the angular position of an object. It's a vector quantity, meaning it has both a magnitude and a direction, typically indicating the axis of rotation and whether it's clockwise or counterclockwise. In physics problems, angular velocity assists in understanding how quickly an object is rotating.

The relationship between angular velocity and linear velocity provides a bridge between linear and rotational motion. One key point is that while linear speed can differ across points on a spinning object, all points share the same angular velocity. Recognizing this allows students to accurately describe the motion of different parts of a rotating system and is pivotal in the analysis of rotational dynamics.

Rotational Kinematics

Rotational kinematics involves the description of rotational motion without considering the forces that cause the motion. It is analogous to linear kinematics but for rotational motion. Key quantities include rotational displacement (theta), angular velocity (omega), and angular acceleration (alpha). The main equations of rotational kinematics closely mirror those of linear motion, but with angular counterparts.

For example, just as displacement, velocity, and acceleration describe linear motion, similar formulas apply to angular displacement, angular velocity, and angular acceleration in rotation. Understanding how these quantities relate to each other in rotational kinematics is instrumental in solving problems involving spinning objects, from simple wheels to complex planetary systems. Mastery of these concepts enables students to approach a variety of physics challenges with confidence, further enhancing their grasp of mechanics as a whole.