At a bulk loading station, gravel leaves the hopper at the...

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of $10 \mathrm{ft} / \mathrm{sec}$ in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is $380 \mathrm{lb}$, which overcomes the 200 lb of frictional road resistance. Determine the acceleration $a$ of the truck 4 seconds after the hopper is opened over the truck bed, at which instant the truck has a forward speed of $1.5 \mathrm{mi} / \mathrm{hr}$ The empty weight of the truck is $12,000 \mathrm{lb}$.

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of $10 \mathrm{ft} / \mathrm{sec}$ in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is $380 \mathrm{lb}$, which overcomes the 200 lb of frictional road resistance. Determine the acceleration $a$ of the truck 4 seconds after
the hopper is opened over the truck bed, at which instant the truck has a forward speed of $1.5 \mathrm{mi} / \mathrm{hr}$ The empty weight of the truck is $12,000 \mathrm{lb}$.

Key Concepts

Variable Mass Systems

This concept involves systems whose mass changes over time. In many real-world problems, such as a truck that gains load, the mass isn’t constant, and the analysis must account for the additional mass being added. This requires setting up momentum balances that reflect both the changes in velocity and the changes in mass, leading to a modified form of Newton's second law.

Conservation of Momentum

Conservation of momentum is fundamental when analyzing systems where mass is added or removed. In a variable mass system, momentum conservation principles are used to relate the incoming momentum of an added mass to the momentum of the system as a whole. This concept helps in deriving the equations needed to find acceleration when external forces are present.

Force Analysis in Mechanics

Force analysis is key in understanding the net forces acting on a moving body. The problem involves combining the driving force, frictional resistance, and the effects of additional mass loading. A careful balance of these forces, often through a free-body diagram, is necessary to determine acceleration at a given time.

Momentum Flux

Momentum flux refers to the rate at which momentum enters or leaves a system due to a mass flow. In the context of the truck gaining weight from gravel, the momentum brought in by the supplemental mass contributes to the overall momentum balance, and this must be taken into account to properly compute the acceleration of the truck.

Units Conversion in Dynamics

Units conversion is critical when dealing with dynamic problems, especially when quantities like mass, force, and velocity are given in different systems. Proper conversions ensure consistency in the equations of motion, so that computed values like acceleration are accurate and meaningful within the chosen system of units.

Transcript

Please give Ace some feedback