A 2.0 kg pendulum bob swings on a 2.0-m-long string. The bob's speed is 1.5 m/s when the string makes a 15° angle with vertical and the bob is moving toward the bottom of the arc. At this instant, what are the magnitudes of the tension in the string?

Verified step by step guidance

Step 1: Identify the forces acting on the pendulum bob. These include the tension in the string (T) and the gravitational force (mg). The tension has both vertical and horizontal components, while gravity acts vertically downward.

Step 2: Break the forces into components. The gravitational force is given by \( F_g = m \cdot g \), where \( m = 2.0 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). The tension force has a vertical component \( T \cdot \cos(\theta) \) and a horizontal component \( T \cdot \sin(\theta) \), where \( \theta = 15^\circ \).

Step 3: Use Newton's second law in the radial direction to relate the forces to the centripetal acceleration. The centripetal force is provided by the horizontal component of the tension and is given by \( F_c = \frac{m \cdot v^2}{r} \), where \( v = 1.5 \, \text{m/s} \) and \( r = 2.0 \, \text{m} \).

Step 4: Write the equation for the vertical forces. The vertical component of the tension must balance the gravitational force: \( T \cdot \cos(\theta) = m \cdot g \). Solve for \( T \) using this equation.

Step 5: Combine the equations for the vertical and radial forces to find the total tension \( T \). Substitute the known values for \( m \), \( g \), \( v \), \( r \), and \( \theta \) into the equations to calculate \( T \).

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Key Concepts

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

Tension in a String

Tension is the force exerted by a string or rope when it is pulled tight by forces acting from opposite ends. In the context of a pendulum, tension acts along the string and counteracts the weight of the bob while also providing the necessary centripetal force to keep the bob moving in a circular path.

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. For a pendulum, this force is provided by the tension in the string and is essential for maintaining the circular motion of the bob as it swings.

Forces Acting on the Pendulum Bob

At any point in its swing, the pendulum bob experiences two main forces: gravitational force acting downward and tension in the string acting upward along the string. The net force acting on the bob determines its acceleration and is crucial for calculating the tension when the bob is at an angle to the vertical.

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