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) Acceleration: 4.64 m/s² upward; (b) Tension: 104.91 N.

Step 1: Analyzing Forces

Begin by identifying the forces acting on the ball at its lowest point: the gravitational force \( F_g \) and the tension \( T \) in the rope. The gravitational force acts downward and is equal to the weight of the ball, \( F_g = 71.2 \; \text{N} \). The tension force \( T \) acts along the rope, in the vertical direction.

Step 2: Applying Newton's Second Law

At the lowest point, the ball undergoes centripetal acceleration directed towards the center of the circular path. Hence, apply Newton's Second Law in the radial direction: \[ T - F_g = m v^2 / r, \] where \( m \) is the mass of the bowling ball, \( v \) is the velocity \( 4.20 \; \mathrm{m/s} \), and \( r \) is the rope length \( 3.80 \; \mathrm{m} \). First, find the mass from: \( F_g = mg \), \( m = \frac{F_g}{g} = \frac{71.2}{9.81} \; \mathrm{kg}. \)

Step 3: Calculating Mass of the Ball

Compute the mass: \[ m = \frac{71.2}{9.81} \approx 7.26 \; \mathrm{kg}. \] Use this mass to further analyze the forces in subsequent steps.

Step 4: Solving for Centripetal Acceleration

Find the centripetal acceleration \( a_c \) using: \[ a_c = \frac{v^2}{r} = \frac{(4.20)^2}{3.80} = \frac{17.64}{3.80} \approx 4.64 \; \mathrm{m/s^2}. \] The direction of this acceleration is towards the center of the circular path, which is upward along the rope direction.

Step 5: Calculating Tension in the Rope

Determine the tension \( T \) using: \[ T = F_g + m a_c = 71.2 + 7.26 \times 4.64. \] Compute: \[ T = 71.2 + 33.71 \approx 104.91 \; \mathrm{N}. \] Thus, the tension in the rope is approximately \( 104.91 \; \mathrm{N}. \)

Key Concepts

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle that describes the relationship between an object's mass, the force acting upon it, and its acceleration. This law is elegantly summarized by the formula: \[ F = ma \] where \( F \) is the net force acting on the object, \( m \) is the mass of the object, and \( a \) is the acceleration of the object. This equation means that an object's acceleration is directly proportional to the net force acting on it and inversely proportional to its mass.

• When multiple forces act on an object, the net force is the vector sum of all these forces.
• In our pendulum exercise, we have two main forces to consider: gravitational force acting downwards and the tension in the rope acting upwards.
• At the lowest point of the swing, the acceleration is centripetal, pointing towards the center of the circular path formed by the pendulum’s motion.

It is crucial to apply Newton's Second Law for understanding how these forces contribute to the net force that produces centripetal acceleration. The law helps in calculating unknown forces, such as the tension in the rope, by setting up equations that balance the forces in motion.

Centripetal Acceleration

Centripetal acceleration is the acceleration that occurs in objects moving in circular paths. It acts towards the center of the circle and keeps the object in a rotational trajectory. The formula for centripetal acceleration is: \[ a_c = \frac{v^2}{r} \] where \( v \) is the speed of the object and \( r \) is the radius of the circular path.

• In the pendulum motion, the radius \( r \) is the length of the rope, and the object is the bowling ball.
• The centripetal force necessary to maintain this motion is provided by the tension in the rope.
• At the lowest point, where the speed is highest, the acceleration is maximum, pointing towards the pivot point at the ceiling.

Understanding centripetal acceleration is key in analyzing circular motion dynamics, such as in a pendulum, where gravity and tension work together to sustain the swing. Calculating this acceleration helps determine the necessary tension in the rope, ensuring the pendulum continues its regular path.

Free-Body Diagram

A free-body diagram is a visual representation used to illustrate all the forces acting on an object. It is a crucial tool for solving physics problems, as it helps to break down complex interactions into manageable components.

• This diagram typically includes vectors to represent forces like gravity, tension, normal force, and applied forces.
• In the pendulum scenario, the free-body diagram shows two main forces: the gravitational force acting downward and the tension in the rope pulling upward towards the pivot.
• Understanding the direction and magnitude of each force allows for better application of physical laws like Newton's second law.

By constructing a free-body diagram, one can visualize force interactions and apply the necessary equations to solve for unknowns, such as the precise tension in the rope or the direction of acceleration. It makes abstract forces tangible and easier to analyze for practical physics problems.