Problem 138
Question
The shoulder joint can rotate at about 25 radians per second. Assuming that a golfer's arm is straight and the distance from the shoulder to the clubhead is 5 feet, approximate the linear speed of the clubhead from the shoulder rotation.
Step-by-Step Solution
Verified Answer
The linear speed of the clubhead is 125 feet per second.
1Step 1: Understand the Relationship Between Angular and Linear Speed
The problem asks for the linear speed of the golf club's head due to the rotation at the shoulder. The formula for linear speed \( v \) of an object moving in a circular path is given by \( v = r \cdot \omega \) where \( r \) is the radius (or distance from the rotation point) and \( \omega \) is the angular speed.
2Step 2: Identify the Values
We are given that the angular speed \( \omega \) of the shoulder is 25 radians per second, and the distance \( r \) from the shoulder to the clubhead is 5 feet.
3Step 3: Calculate the Linear Speed
Using the formula \( v = r \cdot \omega \), substitute the known values: \( r = 5 \text{ feet} \) and \( \omega = 25 \text{ radians per second} \). Calculating the linear speed, we find \( v = 5 \times 25 = 125 \text{ feet per second} \).
Key Concepts
Linear SpeedCircular MotionRadians per Second
Linear Speed
Linear speed refers to how fast an object moves along a straight path. In the context of circular motion, such as the rotation of a golf club, it describes how quickly a point on the rotating object (like the clubhead) travels along the circle's edge. This speed can vary depending on how far the point is from the center of rotation.
Linear speed is crucial because it helps us understand the motion dynamics of rotating bodies. For example, in our problem, we calculate how fast the clubhead moves as the golfer swings by using the shoulder as the pivot point. We can determine this by multiplying the angular speed by the distance from the shoulder to the clubhead.
To sum up, linear speed gives us insight into how fast something is traveling along its path, especially in rotational motions where different points on the rotating object travel at different speeds.
Linear speed is crucial because it helps us understand the motion dynamics of rotating bodies. For example, in our problem, we calculate how fast the clubhead moves as the golfer swings by using the shoulder as the pivot point. We can determine this by multiplying the angular speed by the distance from the shoulder to the clubhead.
To sum up, linear speed gives us insight into how fast something is traveling along its path, especially in rotational motions where different points on the rotating object travel at different speeds.
Circular Motion
Circular motion occurs when an object moves along a circular path. This type of motion is all around us, from planets orbiting stars to wheels turning on vehicles.
When something rotates or revolves around a central point or axis, it is said to be in circular motion. The rotation can be at a constant speed, meaning each part of the object completes a circle at the same rate. In our case, the golfer's clubhead travels around the shoulder joint, creating a circular path as they swing.
Key concepts related to circular motion include:
When something rotates or revolves around a central point or axis, it is said to be in circular motion. The rotation can be at a constant speed, meaning each part of the object completes a circle at the same rate. In our case, the golfer's clubhead travels around the shoulder joint, creating a circular path as they swing.
Key concepts related to circular motion include:
- **Radius of the circle:** This is the distance from the center of the circle to any point on its edge. For a rotating object, this would be from the pivot point to the rotating point.
- **Angular speed:** This describes how fast an object rotates, typically measured in radians per second. It indicates how many radians the object covers within a certain time period.
Radians per Second
Radians per second is a unit of angular speed. It measures how quickly an angle changes in circular motion, with one radian representing the angle made when the arc length is equal to the circle's radius.
In the given problem, we know that the shoulder rotates at 25 radians per second. This unit tells us how rapidly the shoulder's position changes around the joint, influencing the speed of the golf club's swing.
To connect this to real-world applications, consider how a rapidly spinning wheel has a high angular speed measured in radians per second. The wheel covers many radians in a short time frame, suggesting rapid rotation.
In the given problem, we know that the shoulder rotates at 25 radians per second. This unit tells us how rapidly the shoulder's position changes around the joint, influencing the speed of the golf club's swing.
To connect this to real-world applications, consider how a rapidly spinning wheel has a high angular speed measured in radians per second. The wheel covers many radians in a short time frame, suggesting rapid rotation.
- **Practical importance:** Knowing angular speed allows us to calculate the linear speed if the radius is known, crucial for determining how fast a rotating section travels.
- **Conversion utility:** Radians offer a straightforward way to link angular speed to other motion measures, like linear speed, given that 1 radian is approximately 57.3 degrees.
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