Problem 2 - University Physics with Modern Physics

Question

A \(5.00-\mathrm{kg}\) sack of flour is lifted vertically at a constant speed of 3.50 \(\mathrm{m} / \mathrm{s}\) through a height of 15.0 \(\mathrm{m}\) . (a) How great a force is required? (b) How much work is done on the sack by the lifting force? What becomes of this work?

Step-by-Step Solution

Answer

(a) 49.05 N; (b) 735.75 J. The work becomes potential energy.

Step 1: Understand the Problem

We need to find the force required to lift a sack of flour at a constant speed and calculate the work done by this force. The constant speed implies that the net force on the sack is zero since there is no acceleration.

Step 2: Calculate the Force Required

The force needed must balance the gravitational force acting on the sack for it to move at a constant speed. This means the lifting force should equal the weight of the sack. Calculate this using: \[ F = m \cdot g \]where \( m = 5.00 \; \text{kg} \) and \( g = 9.81 \; \text{m/s}^2 \).\[ F = 5.00 \; \text{kg} \times 9.81 \; \text{m/s}^2 = 49.05 \; \text{N} \]

Step 3: Calculate the Work Done

Work done by the lifting force on the sack when moving it through a height is calculated as: \[ W = F \cdot d \cdot \cos(\theta) \]where \( d = 15.0 \; \text{m} \) and \( \theta = 0 \text{ degrees} \) since the force and movement are in the same direction. \[ W = 49.05 \; \text{N} \times 15.0 \; \text{m} \times \cos(0) = 735.75 \; \text{J} \]

Step 4: Interpret the Result

The work done (735.75 J) goes into increasing the potential energy of the sack due to its increased height. This stored potential energy could be transformed into other forms if the sack falls or is lowered at a later time.

Key Concepts

Newton's laws of motion

Isaac Newton was an influential scientist, and his laws of motion are fundamental principles in physics. These laws explain how objects behave when subjected to external forces. Here's a quick rundown:

• **First Law (Law of Inertia):** An object will remain at rest or move at a constant velocity unless acted upon by an external force.
• **Second Law (Law of Acceleration):** The rate of change of momentum of an object is proportional to the applied force and occurs in the direction of the force. Mathematically, it is expressed as \[ F = m imes a \]
• **Third Law (Action and Reaction):** For every action, there is an equal and opposite reaction.

In our exercise, Newton's second law is particularly relevant. Since the sack of flour is moving at a constant speed, it indicates a net force of zero. This means the upward lifting force exactly balances the downward gravitational force, according to Newton's second law.

gravitational force

Gravitational force is a force of attraction that acts between any two masses. It is a universal force acting between all matter. On Earth, this force gives weight to physical objects and causes them to fall towards the ground when dropped. The gravitational force on an object can be calculated as:\[ F_{ ext{gravity}} = m imes g \]where:

• \( m \) is the mass of the object (in kilograms)
• \( g \) is the acceleration due to gravity, on average \( 9.81 \text{ m/s}^2 \) on Earth

In our case, the gravitational force (or weight) acting on the 5 kg sack of flour is:\( \ 49.05 \ \text{N} \). This force must be overcome by the lifting force to keep the sack moving upwards at a constant speed.

potential energy

Potential energy represents the stored energy of an object due to its position or state. When an object is lifted against gravity, it gains gravitational potential energy, which depends on the height and mass of the object as well as the strength of gravity. The formula for gravitational potential energy is:\[ PE = m imes g imes h \]where:

• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity
• \( h \) is the height the object is lifted to

For our sack of flour, the potential energy gained when lifted to a height of 15 meters is:\[ 735.75 \, \text{Joules} \]. This energy is stored and can be converted back into kinetic energy if the sack were to fall.

work done calculation

Work done is a measure of energy transfer when a force moves an object over a distance. It is calculated using the equation:\[ W = F imes d imes \cos(\theta) \]where:

• \( F \) is the force applied (in newtons)
• \( d \) is the displacement of the object (in meters)
• \( \theta \) is the angle between the force and the direction of movement

In our scenario, the force and displacement are in the same direction (upwards), making \( \theta = 0 \) degrees and \( \cos(\theta) = 1 \). The work done to lift the sack of flour is thus:\[ 735.75 \, \text{Joules} \]. This amount of work is transferred into potential energy, as the height increases, preparing it for any subsequent motion like falling back down.