Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 26

Chapter 8, Problem 26

A heavy ball with a weight of 100 N (m = 10.2 kg) is hung from the ceiling of a lecture hall on a 4.5-m-long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.5 m/s as it passes through the lowest point. What is the tension in the rope at that point?

Verified step by step guidance

Step 1: Identify the forces acting on the ball at the lowest point of the swing. These include the gravitational force (weight) acting downward and the tension in the rope acting upward. Additionally, the ball experiences a centripetal force due to its circular motion.

Step 2: Write the equation for the net force in the vertical direction at the lowest point. The net force is the centripetal force, which is provided by the difference between the tension in the rope and the gravitational force. Mathematically, this can be expressed as:

T - W = \frac{m}{v} r

, where T is the tension, W is the weight, m is the mass, v is the speed, and r is the radius of the circular motion (equal to the length of the rope).

Step 3: Rearrange the equation to solve for the tension T. This gives:

T = W + \frac{m}{v} r

. Substitute the known values: W = 100 N, m = 10.2 kg, v = 5.5 m/s, and r = 4.5 m.

Step 4: Calculate the centripetal force term

\frac{m}{v} r

. This involves squaring the speed (v^2), multiplying by the mass (m), and dividing by the radius (r).

Step 5: Add the gravitational force (W) to the centripetal force term to find the total tension (T) in the rope. Ensure that the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for analyzing forces acting on the pendulum at the lowest point of its swing, where both gravitational and tension forces are at play.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In the case of the pendulum, as the ball swings through the lowest point, the tension in the rope provides the necessary centripetal force to maintain its circular motion.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. As the pendulum swings, potential energy at the highest point converts to kinetic energy at the lowest point, allowing us to analyze the forces acting on the ball and calculate the tension in the rope.

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