Problem 21 - College Physics

Question

A student weighing \(800 \mathrm{~N}\) crouches on a scale and suddenly springs vertically upward. His roommate notices that the scale reads 900 N momentarily just as he leaves the scale. With what acceleration does he leave the scale?

Step-by-Step Solution

Answer

The student leaves the scale with an acceleration of approximately 1.225 m/s².

Step 1: Understand the Forces Involved

The student weighs 800 N, which is the force due to gravity acting downward. When the scale reads 900 N, this is the normal force exerted by the scale on the student as he jumps upward.

Step 2: Apply Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the mass of the object times its acceleration: \[ F_{net} = m \cdot a \] Where \( F_{net} = F_{normal} - F_{gravity} \).

Step 3: Calculate the Net Force

The net force \( F_{net} \) is calculated as:\[ F_{net} = 900 \, \text{N} - 800 \, \text{N} = 100 \, \text{N} \]

Step 4: Determine the Mass of the Student

The weight of the student is the force of gravity acting on him, given by:\[ F_{gravity} = m \cdot g \] Where \( g \approx 9.8 \, \text{m/s}^2 \). Therefore, the mass \( m \) is:\[ m = \frac{F_{gravity}}{g} = \frac{800}{9.8} \approx 81.63 \, \text{kg} \]

Step 5: Solve for Acceleration

Using the equation from Step 2, substitute the net force and mass values:\[ 100 = 81.63 \cdot a \] So, acceleration \( a \) is:\[ a = \frac{100}{81.63} \approx 1.225 \, \text{m/s}^2 \]

Key Concepts

Newton's Second Law

One of the fundamental principles in physics, Newton's Second Law, gives us a way to relate force, mass, and acceleration. It states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This can be expressed with the formula:

• \( F_{net} = m \cdot a \)

This equation helps us understand how changes in the forces acting on an object affect its motion. For instance, if the mass is constant, increasing the force will increase the acceleration.
Similarly, for a constant force, a greater mass means less acceleration. This law is crucial because it provides a mathematical methodology to dissect and predict the behavior of moving objects in various situations.

Net Force

The net force is the overall force acting on an object after all individual forces are considered. It is the key factor that causes a change in the motion of an object according to Newton's Second Law.

• In the example of the student on the scale, while standing, he exerts a downward force equal to his weight, which is 800 N.
• During the jump, the scale shows 900 N, indicating the normal force exerted upwards by the scale is affected by the action of jumping.
• The net force is what remains once these forces are considered together: \( F_{net} = F_{normal} - F_{gravity} = 900 \, \text{N} - 800 \, \text{N} = 100 \, \text{N} \).

Understanding net force is essential to investigating why objects speed up or slow down, as it dictates the overall effect of all forces acting on the object.

Normal Force

The normal force is a support force exerted upon an object that is in contact with another stable object. It acts perpendicular to the surface of contact and opposes the force of gravity.

• In the jumping student scenario, when the student crouches and prepares to jump, the scale provides a normal force to counteract gravity and even more when he pushes off.
• It's the increase in normal force as he jumps that allows for the lift-off.
• The momentary reading of 900 N on the scale indicates this increased force compared to the regular reading of 800 N when he is just standing still.

The normal force can vary significantly depending on the situation, such as when jumping or sitting on an incline, and is crucial for understanding contact interactions.

Acceleration Calculation

Acceleration is a measure of how quickly an object's speed or direction is changing. Using the given scenario, we use the net force and mass to find acceleration by rearranging Newton's Second Law:

• First, calculate the mass from the weight using gravity: \( m = \frac{800}{9.8} \approx 81.63 \, \text{kg} \).
• Then solve for acceleration: \( a = \frac{F_{net}}{m} = \frac{100}{81.63} \approx 1.225 \, \text{m/s}^2 \).

This shows how knowing the net force acting on an object and its mass can determine how fast it will accelerate. Understanding this process is vital, as it underpins motion dynamics and is foundational for solving many real-world physics problems involving movement.