A pendulum is made by tying a 500 g ball to a 75-cm-long string. The pendulum is pulled 30° to one side, then released. What is the ball's speed at the lowest point of its trajectory?

Verified step by step guidance

1. Step 1: Identify the type of energy transformation involved. The pendulum converts potential energy at its highest point (when it is displaced by 30 degrees) into kinetic energy at the lowest point of its trajectory.
2. Step 2: Write the expression for gravitational potential energy at the highest point: \( U = m g h \), where \( m \) is the mass of the ball (0.500 kg), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height of the ball above the lowest point.
3. Step 3: Determine the height \( h \) using trigonometry. The string length is \( L = 0.75 \) m, and the pendulum is displaced by 30 degrees. The vertical height \( h \) can be calculated as \( h = L - L \cos(\theta) \), where \( \theta = 30 \) degrees.
4. Step 4: At the lowest point, all the potential energy is converted into kinetic energy. Write the expression for kinetic energy: \( K = \frac{1}{2} m v^2 \), where \( v \) is the speed of the ball at the lowest point. Set the potential energy equal to the kinetic energy: \( m g h = \frac{1}{2} m v^2 \).
5. Step 5: Solve for \( v \) by canceling \( m \) from both sides and rearranging the equation: \( v = \sqrt{2 g h} \). Substitute the values of \( g \) and \( h \) (calculated in Step 3) to find the speed of the ball at the lowest point.

Key Concepts

Conservation of Energy

The principle of conservation of energy states that in a closed system, the total energy remains constant. In the case of the pendulum, the potential energy at the highest point (when pulled to the side) is converted into kinetic energy at the lowest point. This relationship allows us to calculate the speed of the ball at the lowest point using the initial height and mass.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For the pendulum, when the ball is raised to a certain height, it gains gravitational potential energy, which can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above the lowest point.

Kinetic Energy

Kinetic energy is the energy of an object in motion, defined by the formula KE = 1/2 mv², where m is mass and v is velocity. At the lowest point of the pendulum's swing, all the potential energy has been converted into kinetic energy, allowing us to determine the speed of the ball by equating the potential energy at the starting height to the kinetic energy at the lowest point.