Problem 42 - University Physics with Modern Physics

Question

A parachutist relies on air resistance (mainly on her parachute) to decrease her downward velocity. She and her parachute have a mass of \(55.0 \mathrm{kg},\) and air resistance exerts a total upward force of 620 \(\mathrm{N}\) on her and her parachute. (a) What is the weight of the parachutist? (b) Draw a free-body diagram for the parachutist (see Section 4.6 ). Use that diagram to calculate the net force on the parachutist. Is the net force upward or downward? (c) What is the acceleration (magnitude and direction) of the parachutist?

Step-by-Step Solution

Answer

(a) The weight is 539 N. (b) Net force is 81 N upward. (c) Acceleration is 1.47 m/s² upward.

Step 1: Calculate the Weight of the Parachutist

The weight of an object can be calculated using the formula: \( W = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)). Given that the mass \( m = 55.0 \, \text{kg} \), the weight \( W \) is calculated as follows:\[ W = 55.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 539 \, \text{N} \]

Step 2: Draw a Free-Body Diagram and Analyze Forces

In a free-body diagram, you represent all the forces acting on the parachutist.1. **Weight** \( (W) \): Acts downward, calculated as **539 N**.2. **Air Resistance** \( (F_r) \): Acts upward, given as **620 N**.The net force \( (F_{net}) \) acting on the parachutist is the difference between the upward force and the downward force:\[ F_{net} = F_r - W = 620 \, \text{N} - 539 \, \text{N} = 81 \, \text{N} \]The net force is upward because the air resistance force is greater than the weight.

Step 3: Calculate the Acceleration of the Parachutist

The acceleration \( (a) \) can be calculated using Newton's second law \( F = ma \), where \( F \) is the net force and \( a \) is the acceleration.Rearranging the formula to solve for \( a \):\[ a = \frac{F_{net}}{m} = \frac{81 \, \text{N}}{55.0 \, \text{kg}} \approx 1.47 \, \text{m/s}^2 \]The parachutist is accelerating upward at a rate of approximately \( 1.47 \, \text{m/s}^2 \).

Key Concepts

Understanding Free-Body Diagrams

When dealing with forces and motion, it's helpful to draw a free-body diagram. This diagram is a simple representation showing all the forces acting on an object. In the case of the parachutist, two main forces are at play:

• **Weight**: This force pulls the parachutist downward due to gravity. It can be calculated using the formula: \( W = mg \), where \( m \) is mass and \( g \) is gravity (approx. \( 9.8 \, \text{m/s}^2 \)). For the parachutist, this amounts to 539 N downward.
• **Air Resistance**: This is the force that pushes up against the falling parachutist and slows her descent. In this exercise, it is given as 620 N upward.

By placing these forces on a free-body diagram, you can visually assess the net force. The length and direction of arrows in the diagram correspond to the magnitude and direction of these forces.
Making a clean diagram helps in understanding which forces are stronger, and the overall effect they have on the object's motion.

Exploring Air Resistance

Air resistance is a type of frictional force that acts against the motion of an object through the air. For a parachutist, this force plays a crucial role in making descents slower and safer. It mainly acts on the parachute, providing significant upward force against gravity.
Air resistance depends on several factors:

• **Velocity of the object**: The faster the object falls, the greater the air resistance.
• **Surface area**: A bigger parachute catches more air and increases the resistance.
• **Shape of the object**: Streamlined shapes encounter less resistance compared to bulky ones.

In the exercise, air resistance is strong enough to create a net upward force when the upward force exceeds the gravitational pull, allowing the parachutist to descend slowly.
Understanding how air resistance compensates for gravity helps in designing better parachutes and calculating safe descent speeds.

Calculating Net Force

The concept of net force is crucial in understanding motion as described by Newton's Second Law. To find the net force, you must take into account all the forces acting on the object and determine the overall direction and magnitude.

• In our parachutist example, two main forces are considered: the downward gravitational weight (539 N) and the upward air resistance (620 N).
• The net force is the difference between these two opposing forces: \( F_{net} = F_r - W = 620 \, \text{N} - 539 \, \text{N} = 81 \, \text{N} \).
• Since 620 N (upward) is greater than 539 N (downward), the net force is 81 N upwards.

Once you calculate the net force, you can find the acceleration by using the formula \( F = ma \) rearranged to \( a = \frac{F_{net}}{m} \).
The net force directs the movement of the parachutist, showing us that she is accelerating upward as long as air resistance exceeds her weight.