Question

In a modified tug-of-war game, two people pull in opposite directions, not on a rope, but on a \(25-\mathrm{kg}\) sled resting on an icy road. If the participants exert forces of \(90 \mathrm{~N}\) and \(92 \mathrm{~N}\), what is the acceleration of the sled?

Short answer

The acceleration of the sled is \(0.08 \mathrm{~m/s}^2\).

Step-by-step solution

Identify the given forces

Identify the forces exerted by the two participants. In this case, one force is \(90 \mathrm{~N}\) and the other is \(92 \mathrm{~N}\).

Calculate the net force

Subtract the smaller force from the larger one to get the net force. In this case, net force \( F = 92 \mathrm{~N} - 90 \mathrm{~N} = 2 \mathrm{~N}\).

Identify the mass of the sled

Identify the mass of the sled, which in this case is \(25 \mathrm{~kg}\).

Apply Newton's second law of motion

Use the formula for Newton's second law of motion, which states that acceleration = force / mass. In this case, acceleration (\(a\)) = \(2 \mathrm{~N}/25 \mathrm{~kg}\).

Compute the acceleration

Compute the acceleration by dividing the net force by the mass. We get acceleration (\(a\)) = \(2 \mathrm{~N}/25 \mathrm{~kg} = 0.08 \mathrm{~m/s}^2\).

Key concepts

Net Force Calculation

Understanding the net force calculation is crucial in solving problems related to Newton's second law of motion. The net force is the vector sum of all the forces acting on an object. When several forces are acting on an object in different directions, you must calculate the resultant force that will cause the object to move or change its motion.

To calculate the net force, one should take into account both the magnitude and the direction of each individual force. If two forces are acting in opposite directions, as in a tug-of-war, you subtract the smaller force from the larger one to find the net force. For example, if there are forces of 90 N and 92 N acting on a sled in opposite directions, the net force exerted on the sled is the difference between these two forces, which is 92 N - 90 N = 2 N. This net force is what will create an acceleration in the object.

Force and Acceleration

The concept of force and acceleration is a cornerstone in understanding mechanics. According to Newton's second law, the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass, often expressed as the formula: \( a = \frac{F}{m} \), where \( a \) is the acceleration, \( F \) is the net force, and \( m \) is the mass of the object.

In our sled example, with a net force of 2 N acting upon a 25 kg sled, we use this law to find how quickly the sled will accelerate. Substituting the known values into the formula gives us \( a = \frac{2N}{25kg} \), which after calculation shows the sled will accelerate at 0.08 meters per second squared. It’s important to consider the direction as well; the sled will accelerate in the direction of the net force.

Mass and Acceleration

The relationship between mass and acceleration is an integral part of Newton's second law. While acceleration is a measure of how quickly an object's velocity changes, mass is a measure of the amount of matter in an object. The larger the mass of an object, the more force it takes to change its motion, hence the less acceleration it will have for a given force.

In the scenario provided, the 25 kg sled requires a certain amount of force to change its state of motion. When only a small net force is applied (2 N in this case), the resulting acceleration is small as well, resulting in a gentle increase in the sled's speed. If the sled had less mass or if the net force was greater, the acceleration would be higher. This illustrates how mass and acceleration are inversely related; as one goes up, the other must go down, assuming the net force remains constant.