A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) What is the gravitational force acting on the ball?

검증된 단계별 안내

Convert the mass of the ball from grams to kilograms, as the SI unit for mass is kilograms. Use the conversion: \( 1 \text{ g} = 0.001 \text{ kg} \). Thus, \( m = 500 \text{ g} = 0.5 \text{ kg} \).

Recall the formula for gravitational force: \( F_g = m \cdot g \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth's surface).

Substitute the values into the formula: \( F_g = 0.5 \cdot 9.8 \).

Simplify the expression to calculate the gravitational force acting on the ball. This will give the force in newtons (N).

The gravitational force calculated represents the constant downward force acting on the ball due to Earth's gravity, regardless of its position in the vertical circle.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. For an object near the Earth's surface, this force can be calculated using the formula F = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.81 m/s²). In this scenario, the gravitational force acting on the ball can be determined by multiplying its mass (0.5 kg) by g.

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. It is essential for maintaining circular motion and can be calculated using the formula F_c = mv²/r, where m is the mass, v is the speed, and r is the radius of the circular path. At different points in the vertical circle, the gravitational force and tension in the string contribute to the centripetal force.

Energy Conservation in Circular Motion

In circular motion, mechanical energy is conserved if only conservative forces (like gravity) are acting. The total mechanical energy at the top and bottom of the circle can be analyzed using the principles of kinetic and potential energy. At the top, the ball has both kinetic energy (due to its speed) and potential energy (due to its height), while at the bottom, the potential energy is lower, and the kinetic energy is higher, allowing for the calculation of forces and speeds at different points.