To keep a conveyor belt moving when it transports luggage requires a greater driving force than for an empty belt. What additional driving force is needed if the belt moves at a constant speed of 1.5 m/s and the rate at which luggage is placed on one end of the belt and removed at the other end is 20 kg/s? Assume that the luggage is dropped vertically onto the belt; persons removing luggage grab hold of it and bring it to rest relative to themselves before lifting it off the belt.

To determine the additional driving force required to maintain the conveyor belt's constant speed while transporting luggage, we need to analyze the situation using the principles of momentum and force. The key factor here is that the luggage, when placed on the belt, contributes to the overall momentum that the belt must accommodate to keep moving at a steady speed of 1.5 m/s.

Understanding the Problem

When luggage is placed onto the conveyor belt, it effectively adds mass to the system. The belt must exert enough force to not only keep moving but also to maintain that motion despite the added weight of the luggage. Here’s how we can break down the additional driving force required:

Calculating the Change in Momentum

The rate at which luggage is added to the belt is given as 20 kg/s. Since the speed of the belt is constant at 1.5 m/s, we can calculate the momentum change:

• Momentum (p) is defined as the product of mass (m) and velocity (v): p = m × v.
• Here, the mass flow rate (dm/dt) of luggage is 20 kg/s.
• The velocity (v) of the belt is 1.5 m/s.

Thus, the rate of change of momentum (dp/dt) when luggage is added can be calculated as follows:

dp/dt = (dm/dt) × v = 20 kg/s × 1.5 m/s = 30 kg·m/s².

Relating Momentum Change to Force

According to Newton's second law, the force required to change momentum is equal to the rate of change of momentum. Therefore, the additional force needed to keep the conveyor belt moving at a constant speed while accommodating the luggage is:

F = dp/dt = 30 N.

Conclusion on the Required Driving Force

In simple terms, the conveyor belt needs an additional driving force of 30 Newtons to counteract the momentum change caused by the luggage being placed on it. This force ensures that the belt can maintain its speed of 1.5 m/s without slowing down due to the added load of the luggage.

Real-World Implications

This principle is crucial in designing conveyor systems, as engineers must ensure that motors and drives are capable of providing sufficient torque to overcome not just friction and inertia but also the dynamic forces introduced by loads being transported. Understanding this relationship between mass, speed, and force is vital for efficient conveyor design and operation.