Question

The greenskeepers of golf courses use a stimpmeter to determine how "fast" their greens are. A stimpmeter is a straight aluminum bar with a V-shaped groove on which a golf ball can roll. It is designed to release the golf ball once the angle of the bar with the ground reaches a value of \(\theta=20.0^{\circ} .\) The golf ball \((\) mass \(=1.62 \mathrm{oz}=0.0459 \mathrm{~kg})\) rolls 30.0 in down the bar and then continues to roll along the green for several feet. This distance is called the "reading." The test is done on a level part of the green, and stimpmeter readings between 7 and \(12 \mathrm{ft}\) are considered acceptable. For a stimpmeter reading of \(11.1 \mathrm{ft},\) what is the coefficient of friction between the ball and the green? (The ball is rolling and not sliding, as we usually assume when considering friction, but this does not change the result in this case.)

Short answer

#Answer# To find the coefficient of friction between the golf ball and the green, follow these steps: 1. Determine the acceleration of the golf ball on the stimpmeter using the formula: \(a = g\sin\theta\). 2. Calculate the initial velocity of the golf ball using: \(v = \sqrt{2\cdot g\sin\theta\cdot s}\). 3. Use the conservation of energy to find the work done against friction: \(W = \frac{1}{2}mv^2\). 4. Calculate the friction force with the equation: \(F_friction = \frac{W}{d}\). 5. Find the coefficient of friction using the equation: \(\mu = \frac{F_friction}{F_n}\).

Step-by-step solution

Determine the acceleration of the golf ball on the stimpmeter

Using Newton's second law of motion, the net force acting on the golf ball can be found by multiplying the mass and the acceleration: \(\sum F = ma\). Also, we know the angle of the stimpmeter (\(\theta = 20.0^{\circ}\)), and hence, the acceleration due to gravity (\(g=9.8\,\mathrm{m/s^2}\)) can be divided into components. The component of gravity acting along the inclined plane is \(g\sin\theta\). Therefore, the accelerative force acting on the golf ball along the incline is: \(ma = mg\sin\theta\). The acceleration of the ball can be calculated as: \(a = g\sin\theta\).

Calculate the initial velocity of the golf ball

Now we have the acceleration of the golf ball when it's on the stimpmeter. Next, we need to find its initial velocity right after it leaves the stimpmeter. We'll use the equation of motion for an object with constant acceleration along a straight line: \(v^2 = u^2 + 2as\), where \(v\) is the final velocity (initial velocity right after it leaves the stimpmeter), \(u\) is the initial velocity (0 m/s in this case, as the ball starts from rest), \(a\) is the acceleration, and \(s\) is the distance traveled (30.0 in or 0.762 m). We want to find \(v\), so after rearranging and substituting the known values, we get: \(v = \sqrt{2as} = \sqrt{2\cdot g\sin\theta\cdot s}\).

Use conservation of energy to find the work done against friction

We'll now use the concept of conservation of energy to find the work done against friction. As the ball rolls along the green, it loses kinetic energy, which is transformed into work done against the friction between the ball and the green. The initial kinetic energy of the golf ball is: \(K.E.= \frac{1}{2}mv^2\), where \(m\) is the mass of the golf ball and \(v\) is its initial velocity. The work-energy theorem states: \(W = K.E.\), where \(W\) is the work done against friction.

Calculate the friction force

Now, as we know the work done against friction, we can find the friction force using the following equation: \(W = F_friction\cdot d\), where \(F_friction\) is the friction force and \(d\) is the distance the ball traveled along the green (11.1 feet or 3.38 meters). Rearrange the equation to solve for the friction force: \(F_friction = \frac{W}{d}\).

Find the coefficient of friction

Finally, we can find the coefficient of friction (\(\mu\)) using the equation: \(F_friction = \mu\cdot F_n\), where \(F_n\) is the normal force, which is equal to the gravitational force acting on the golf ball, since there are no other vertical forces acting on it: \(F_n = mg\). Rearrange the equation to solve for the coefficient of friction: \(\mu = \frac{F_friction}{F_n}\). Substitute all values to find the coefficient of friction between the golf ball and the green.

Key concepts

Newton's Second Law of Motion

Newton's Second Law of Motion is a fundamental principle that describes the relationship between an object's motion and the forces acting upon it. In essence, it states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass, which is succinctly expressed as the formula
\( F = ma \).
In the given problem involving a golf ball on a stimpmeter, we apply Newton's second law to calculate the ball's acceleration down the inclined plane. Understanding this law is crucial, as it enables us to analyze the golf ball's motion, considering both the slope's angle and gravity's impact on the ball's acceleration.
When dealing with inclined surfaces, the force of gravity can be broken down into two components: one parallel to the incline, which accelerates the object down the slope, and one perpendicular to the incline, which is counteracted by the normal force. By calculating these components, we can determine the net force and thus the acceleration affecting the golf ball.

Motion on an Inclined Plane

Motion on an inclined plane is slightly more complex than motion on a flat surface, as gravity's impact must be considered in terms of its components along and perpendicular to the surface. For an object on an incline, gravity's force parallel to the surface causes acceleration down the slope, while the perpendicular component contributes to the normal force. This is especially relevant in our golf ball problem. The angle of the incline affects the force applied to the ball, with a steeper angle resulting in a greater component of gravitational force causing acceleration.

Breaking Down Gravity

By knowing the angle \( \theta \) of the stimpmeter and using the component of gravity along the inclined plane \( g\sin\theta \), we find how much of the gravitational force is causing the golf ball to accelerate. This understanding is crucial for predicting the ball's initial velocity as it leaves the ramp, an intermediary step for solving the coefficient of friction.

Work-Energy Theorem

The work-energy theorem is a cornerstone of classical physics, providing a link between the forces acting on an object and its motion. It states that the work done by forces on an object results in a change in its kinetic energy. Therefore, if an object slows down, as in our golf ball example, we can infer that work has been done against it by the friction force of the green.
By calculating the initial kinetic energy of the golf ball right after it leaves the stimpmeter and the work done by the friction force as it rolls to a stop, we can use the work-energy theorem to determine the total work done. This value is crucial for calculating the friction force and ultimately determining the coefficient of friction. The theorem simplifies the problem by allowing us to equate the loss in the ball's kinetic energy directly to the work done by friction.

Conservation of Energy

Conservation of Energy is a principle stating that within a closed system, the total energy remains constant over time. This precept is vital in solving physical problems because it allows us to equate the energy in different forms, knowing that the total must remain constant if no energy enters or leaves the system. In the context of our problem, we use conservation of energy to understand that the golf ball's kinetic energy is not disappearing; instead, it is being transferred into other forms such as heat due to the friction between the ball and the green.
As the golf ball moves across the flat part of the green, the energy it had when leaving the stimpmeter is conserved, barring any external forces. The energy gradually dissipates as the ball does work against the frictional force until it comes to rest. By employing the concept of conservation of energy, we bridge the transition from the ball's dynamic state on the hill to its eventual motionless position, enabling us to calculate the coefficient of friction effectively.