An estimate of the expected load on over-the-shoulder seat...

An estimate of the expected load on over-the-shoulder seat belts is to be made before designing prototype belts that will be evaluated in automobile crash tests. Assuming that an automobile traveling at $45 \mathrm{mi} / \mathrm{h}$ is brought to a stop in $110 \mathrm{~ms},$ determine $(a)$ the average impulsive force exerted by a 200 -lb man on the belt, $(b)$ the maximum force $F_{m}$ exerted on the belt if the force-time diagram has the shape shown.

Key Concepts

Force-Time Diagram Interpretation

A force-time diagram graphically represents how force varies during the duration of an event. The area under the curve of the diagram corresponds to the impulse delivered. Analyzing the shape of the force-time curve allows engineers and physicists to evaluate the distribution of forces, identify maximum force values, and understand the temporal characteristics of the impact, all of which are essential for effective crash safety design.

Impulse-Momentum Theorem

This theorem states that the impulse exerted on an object is equal to its change in momentum. It provides a fundamental connection between force and motion, especially in scenarios where forces act over short time intervals. By integrating the force over the duration of impact, one can determine how the momentum of a mass changes, which is a key concept in crash dynamics and safety analysis.

Average Impulsive Force

The average impulsive force is defined as the constant force that, acting over a given time interval, produces the same impulse as the actual variable force. It is calculated by dividing the total impulse by the time interval over which the impulse is applied. This concept is particularly useful in simplifying the analysis of forces during collisions or rapid deceleration events.

Peak Force in Transient Events

Peak or maximum force is the highest instantaneous force experienced during an event where forces may vary over time. In many real-world scenarios, especially in crash testing, the force-time profile is not uniform, meaning that the maximum force can be substantially higher than the average force. Understanding the peak force is crucial for designing safety equipment that must withstand these transient loads.

Transcript

00:02 In this problem, we have a automobile traveling at a given velocity and is brought to rest in a given time.

00:11 We want to calculate the average impulsive force exerted by a man on the seatbelt and also the maximum force exerted on the seatbelt if the force time diagram is given a certain shape.

00:27 So firstly, let's look at our momentum impulse diagram.

00:32 So the person is moving with the initial momentum mv1 and when the car is brought to rest suddenly there is an impulse acting by the person onto the belt of fdt and the final momentum is zero when the system of the car and the person is brought to rest.

00:58 Now for part a, the force on the belt is opposite to the direction shown.

01:06 The initial velocity we have v1 is 45 miles an hour, which is 66 feet per second.

01:19 And the weight of the person, w, is 200 pounds.

01:27 So if we apply the principle of momentum and impulse, we get that mv1 minus the impulse fdt, is equal to m v2, the final momentum of the system.

01:49 Now ftt, the integral of fdt, is equal to the average force times the change in time, delta t, and delta t we are given.

02:18 So delta t we are given to be 0 .11 seconds.

02:23 So therefore if we go back to our equation, we get that 200 pounds, which is the weight of the man times his velocity 66 feet per second.

02:42 And we divide this by g to get his mass 32 .2 feet per square second minus the average force acting on the individual times the time of 0 .1 .1...

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