Problem 20 - College Physics

Question

A bowling ball (mass \(7.00 \mathrm{~kg}\) and radius \(17.0 \mathrm{~cm}\) ) is released so fast that it skids without rotating down the lane (at least for a while). Assume the ball skids to the right and the coefficient of sliding friction between the ball and the lane surface is \(0.400 .\) (a) What is the direction of the torque exerted by the friction on the ball about the center of mass of the ball? (b) Determine the magnitude of this torque (again about the ball's center of mass).

Step-by-Step Solution

Answer

(a) The torque is directed downward (clockwise from above). (b) The magnitude of the torque is 4.672 Nm.

Step 1: Identify Direction of Torque

The torque direction is determined by the direction of the force (friction) and the radius vector of the ball. Friction acts opposite to the direction of motion, towards the left. Since the ball is skidding to the right, this frictional force creates a clockwise torque on the ball about its center of mass. According to the right-hand rule, if the torque is clockwise, the direction of the torque vector is downward when viewed from above.

Step 2: Calculate Frictional Force

The frictional force can be calculated using the formula: \( F_f = \mu_s \times N \), where \( \mu_s \) is the coefficient of friction, and \( N \) is the normal force. Since the normal force for a ball on a flat surface is equal to its weight, \( N = mg \), plug in the values: \( \mu_s = 0.400 \), \( m = 7.00 \, \mathrm{kg} \), and \( g = 9.81 \, \mathrm{m/s^2} \).

Step 3: Compute Frictional Force

Substitute the given values into the frictional force formula: \( F_f = 0.400 \times (7.00 \, \mathrm{kg} \times 9.81 \, \mathrm{m/s^2}) = 27.48 \, \mathrm{N} \).

Step 4: Calculate Torque Magnitude

The magnitude of the torque due to friction is determined by the formula: \( \tau = r \times F_f \), where \( r \) is the radius of the sphere. Substitute the values: \( r = 0.170 \, \mathrm{m} \) and \( F_f = 27.48 \, \mathrm{N} \).

Step 5: Compute Torque

Evaluate the torque: \( \tau = 0.170 \, \mathrm{m} \times 27.48 \, \mathrm{N} = 4.672 \, \mathrm{Nm} \).

Key Concepts

Frictional Force

Frictional force is a force that opposes motion between two surfaces in contact. In our exercise, this force acts between the bowling ball and the lane as the ball initially skids.
This means it moves without rotating due to insufficient grip between the surfaces. The frictional force always works in the opposite direction of motion, which in this case is to the left, while the ball skids to the right.
Key points to remember about frictional force:

• Always acts parallel to the surfaces in contact.
• Its magnitude depends on the nature of the surfaces and the normal force.
• It is essential in calculating torque, as seen in the calculation of the bowling ball's torque.

This force becomes particularly important in determining rotational effects on objects that move, providing resistance necessary for the ball to eventually start rolling.

Right-Hand Rule

The right-hand rule is a simple way to determine the direction of rotational vectors, such as torque, in physics. It is particularly useful when dealing with rotational motion.
For rotational vectors like torque, you point your right thumb in the direction of the radius (from the center to the point of contact). Your fingers curl in the direction of the applied force, resulting in the thumb pointing in the direction of the resultant vector. In our exercise, as the bowling ball skids right, friction acts to the left. Curling your fingers in the direction of this friction will cause your thumb to naturally point in a direction. This direction corresponds to the vector direction of the torque.

• Helps visualize three-dimensional vector relationships easily.
• Crucial for understanding angular motion and associated forces.

Applying this rule, we see the torque direction as downward from above when the torque is clockwise, assisting in comprehension of complex rotational dynamics.

Coefficient of Friction

The coefficient of friction, often denoted as \( \mu \), is a dimensionless number that characterizes how easily one object can slide over another. It represents the friction between two surfaces.
Since it's dimensionless, it does not have a unit of measurement. This term is pivotal in calculating the force of friction, which directly ties into understanding other phenomena like torque.For our exercise, the coefficient of sliding friction, given as 0.400, helps in determining the frictional force when multiplied by the normal force:

• Frictional force formula: \( F_f = \mu \times N \)
• A higher coefficient means more friction between surfaces, indicating they resist motion more.
• The coefficient of friction can vary based on materials, surface texture, and other conditions.

This concept is essential in determining whether an object will slip, slide, or roll across the surface it moves over.

Rotational Motion

Rotational motion describes an object's movement around a central point, like a pivot or an axis. Understanding this type of motion is key when objects roll or spin instead of just translating linear motion.
In our problem, the initial skid of the bowling ball needs friction to generate torque, eventually making it rotate. This progression from linear skid to rotational motion underlines the relationship between friction, torque, and motion type. Rotational motion concepts include:

• Angular velocity, which measures rotation speed.
• Torque, inducing or altering rotational motion.
• Moment of inertia, analogous to mass in linear motion, indicating resistance to changes in rotation.

Grasping these concepts aids in predicting and understanding real-world movements of everyday objects like wheels and, in this case, a bowling ball. This gives insight into how initial motion is modified into a consistent roll.