Problem 10 - College Physics

Question

A bowling ball weighing 71.2 \(\mathrm{N}\) is attached to the ceiling by a 3.80 \(\mathrm{m}\) rope. The ball is pulled to one side and released; it then swings back and forth like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.20 \(\mathrm{m} / \mathrm{s}\) . At that instant, find (a) the magnitude and direction of the acceleration of the bowling ball and (b) the tension in the rope. Be sure to start with a free-body diagram.

Step-by-Step Solution

Answer

(a) Centripetal acceleration is 4.64 m/s² towards the center. (b) Tension is 104.9 N.

Step 1: Draw the Free-Body Diagram

Identify all forces acting on the bowling ball at the lowest point of the swing. There are two main forces: the gravitational force acting downwards and the tension force in the rope acting upwards and towards the center.

Step 2: Calculate the Centripetal Acceleration

At the lowest point of the swing, the acceleration is purely centripetal. To calculate this, use the formula \( a_c = \frac{v^2}{r} \), where \( v = 4.20 \mathrm{m/s} \) and \( r = 3.80 \mathrm{m} \). Substitute the values to find \( a_c \).\[ a_c = \frac{(4.20)^2}{3.80} = \frac{17.64}{3.80} \approx 4.64 \mathrm{m/s^2} \]

Step 3: Determine the Net Force

The net force acting on the bowling ball when it is at the lowest point is the centripetal force. It can be calculated using \( F_{net} = m \cdot a_c \), where \( m \) is the mass of the bowling ball. First, find \( m \) by using \( m = \frac{W}{g} \), where \( W = 71.2 \mathrm{N} \) and \( g = 9.81 \mathrm{m/s^2} \).

Step 4: Calculate the Mass of the Bowling Ball

Use the formula for weight: \( W = m \cdot g \). Rearrange to find the mass: \( m = \frac{W}{g} = \frac{71.2}{9.81} \). This gives \( m \approx 7.26 \mathrm{kg} \).

Step 5: Calculate the Tension in the Rope

The tension in the rope, \( T \), is the sum of the gravitational force and the centripetal force. Therefore, \( T = W + F_{net} = m \cdot g + m \cdot a_c \). Substitute the values to find the tension. \[ T = 7.26 \cdot 9.81 + 7.26 \cdot 4.64 = 71.2 + 33.7 \approx 104.9 \mathrm{N} \]

Step 6: Report the Magnitude and Direction of Acceleration

The only acceleration at the lowest point is directed towards the center of the circular path. Magnitude of this acceleration is the centripetal acceleration calculated in Step 2, which is \( 4.64 \mathrm{m/s^2} \).

Key Concepts

Centripetal Acceleration in Pendulum Motion

When an object moves in a circular path, it experiences centripetal acceleration. This type of acceleration is always directed towards the center of the circle along which the object is moving. In the case of a pendulum, like the swinging bowling ball, this acceleration is vital because it keeps the ball moving in its curved trajectory.
To calculate centripetal acceleration, use the equation:

• \( a_c = \frac{v^2}{r} \)

where:

• \( v \) is the velocity of the object,
• \( r \) is the radius of the circle.

In our example, at the lowest point of its swing, the velocity of the bowling ball was measured at 4.20 m/s with a rope radius of 3.80 m. This allows us to calculate its centripetal acceleration as approximately 4.64 m/s². Remember, this calculation is crucial for understanding how a pendulum maintains its motion.

Understanding Tension Force in a Pendulum

Tension force is a crucial concept when dealing with swinging pendulums. In a pendulum, the rope or wire’s job is to keep the object in circular motion while providing the necessary support against gravitational pull. As the pendulum swings, the tension varies at different points of the swing. At the swing’s lowest point, the tension is at its highest because it must not only support the object’s weight but also provide for the centripetal force that keeps the object moving in a circle.
To find the tension in the rope at the lowest point of the swing, we calculate the sum of the gravitational force and the net centripetal force needed. The equation used is:

• \( T = W + F_{net} \)
• \( T = m \cdot g + m \cdot a_c \)

where:

• \( T \) is the tension in the rope,
• \( W \) is the gravitational force (weight),
• \( F_{net} \) is the net force,
• \( m \) is the mass,
• \( g \) is the acceleration due to gravity,
• \( a_c \) is the centripetal acceleration.

In this scenario, the calculated tension is approximately 104.9 N.

The Role of a Free-Body Diagram

Free-body diagrams are powerful tools used in physics to analyze forces acting on an object. It's essentially a simple sketch that separates the object from its surroundings and shows all external forces acting upon it. For the pendulum in our problem, the key is to depict forces at the lowest point of its swing, where the analysis is focused.
Two main forces act on the bowling ball:

• The gravitational force, which pulls the ball downward.
• The tension force from the rope, pulling upward and towards the center of its circular path.

By visualizing these forces with a free-body diagram, it becomes easier to set up our equations for calculating other values, such as the tension and the centripetal acceleration. Creating such a diagram enriches your understanding of how these forces interact and resolve together to facilitate the motion, emphasizing the necessary steps for calculating dynamics in pendulum motion.

The Influence of Gravitational Force

Gravitational force is the pull exerted by the Earth on any object with mass, drawing it towards the Earth's center. In pendulum motion, gravity is what pulls the ball downward, ensuring its return swing when it's pulled to one side and released. This force is constant and acts in a vertical direction.
In our example, the force of gravity acting on the bowling ball is equivalent to its weight, calculated as follows:

• \( W = m \cdot g \)

where:

• \( W \) is the weight of the bowling ball,
• \( m \) is the mass,
• \( g = 9.81 \mathrm{m/s^2} \) is the acceleration due to gravity.

The gravitational force calculated in our scenario is 71.2 N. This force plays a crucial role in balancing the tension in the rope during the pendulum's swing, as it is part of the calculations needed to find the total tension force acting at the lowest point of the swing. Understanding gravitational force is key to figuring out how pendulum motions work and how to analyze them using fundamental physics principles.