Newton\'s Laws of Motion Problem Solving

To find the acceleration of an object when a net force is applied, we use Newton's second law of motion, which states that force equals mass times acceleration (F = ma). Rearranging this formula gives us acceleration (a) as a = F/m. Here, the net force (F) is 12 N and the mass (m) is 3 kg. Dividing the force by the mass, we calculate a = 12 N / 3 kg = 4 m/s². Thus, the resulting acceleration of the object is 4 m/s².

To find the acceleration of an object when a net force is applied, we use Newton's second law of motion, which states that force equals mass times acceleration (F = ma). Rearranging the formula gives us acceleration (a) as a = F/m. In this case, a net force of 12 N is applied to a 6 kg object. Plugging in the values, we get a = 12 N / 6 kg = 2 m/s². Thus, the resulting acceleration of the object is 2 m/s².

To find the mass of the object, we can use Newton's second law of motion, which states that force equals mass times acceleration (F = m × a). Rearranging this formula to solve for mass gives us m = F/a. Substituting the given values, we have m = 16 N / 5 m/s², which simplifies to 3.2 kg. Thus, the mass of the object is 3.2 kg, confirming the relationship between force, mass, and acceleration.

To find the force needed to accelerate an object, we use Newton's second law of motion, which states that force equals mass times acceleration (F = m × a). In this case, the skier's mass is 66 kg and the desired acceleration is 1 m/s². Multiplying these values gives F = 66 kg × 1 m/s² = 66 N. Thus, a force of 66 Newtons is required to accelerate the skier at the specified rate.

When an elevator falls freely under the influence of gravity, the only force acting on it is its weight, which can be calculated using the formula \( F = m \cdot g \). Here, \( m \) is the mass of the elevator (1000 kg) and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Multiplying these values gives \( F = 1000 \, \text{kg} \times 9.8 \, \text{m/s}² = 9800 \, \text{N} \). Thus, the force on the falling elevator is 9800 N.

To find the force needed to accelerate a car, we use Newton's second law of motion, which states that force equals mass times acceleration (F = m × a). In this case, the mass of the car is 1400 kg and the desired acceleration is 2 m/s². By multiplying these values (1400 kg × 2 m/s²), we get 2800 N, which is the force required to achieve the specified acceleration.

To calculate the gravitational force acting on an object, we use the formula \( F = m \cdot g \), where \( F \) is the force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity. In this case, the mass \( m \) is 10 kg and \( g \) is 9.8 m/s². Multiplying these values gives \( F = 10 \, \text{kg} \times 9.8 \, \text{m/s}² = 98 \, \text{N} \). Thus, the gravitational force on the object is 98 N.

According to Newton's second law of motion, acceleration is directly proportional to the net force applied to an object when mass is constant. This relationship is expressed as \( F = ma \). If the net force is doubled, the acceleration must also double to maintain the equality, since the mass does not change. Therefore, when the force increases, the acceleration increases proportionally, resulting in double the acceleration.