Question

A worker pushed a \(58.7\) -lb block \((m=26.6 \mathrm{~kg})\) a distance of \(31.3 \mathrm{ft}(=9.54 \mathrm{~m})\) along a level floor at constant speed with a force directed \(32.0^{\circ}\) below the horizontal. The coefficient of kinetic friction is \(0.21\). How much work did the worker do on the block?

Short answer

The worker did 1163.959 work on the block, in units of lb \cdot ft.

Step-by-step solution

Calculate the horizontal component of the force.

This can be done using the formula for the x-component of a force, which is given by \( F_{x} = F \cdot \cos(\Theta) \). Here, \( F \) is the magnitude of the force and \( \Theta \) is the angle the force makes with the horizontal direction. Plugging in the values, we get \( F_{x} = 58.7 \lb \cdot \cos(32.0^{\circ}) = 49.53 \lb \).

Calculate the force of friction.

The force of friction can be calculated using the formula \( F_{f} = \mu \cdot F_{N} \), where \( \mu \) is the coefficient of friction and \( F_{N} \) is the normal force. In this situation, since the block is moving along a level surface, the normal force is just the weight of the block. Therefore, \( F_{f} = 0.21 \cdot 58.7 \lb = 12.327 \lb \).

Calculate the net force applied by the worker.

Since the block moves at a constant speed, the horizontal component of the force applied by the worker is equal to the force of friction. So, the net force \( F_{net} = F_{x} - F_{f} = 49.53 \lb - 12.327 \lb = 37.203 \lb \).

Calculate the work done.

The work done by a force is given by \( W = F_{net} \cdot d \), where \( d \) is the distance traveled. Substituting the given values, we find that \( W = 37.203 \lb \cdot 31.3 ft = 1163.959 lb \cdot ft \).

Key concepts

Force Components

When a force is applied at an angle, it can be broken down into components that are easier to work with. In physics, these components are typically the horizontal and vertical components. This is important because it helps us understand the effect of the force in each direction:

• The horizontal component is found using the formula \( F_{x} = F \cdot \cos(\Theta) \), where \( F \) is the total force and \( \Theta \) is the angle with respect to the horizontal.
• The vertical component uses \( F_{y} = F \cdot \sin(\Theta) \), though it's often not needed if we are only interested in horizontal motion.

For example, if a force is pushing downward at an angle while moving an object, the horizontal component will show how much of that force is actually helping to move the object forward. Figuring out these components is crucial, as it allows us to focus on just the parts of the force that are relevant to our calculations.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It acts in the opposite direction of motion and is calculated with the formula:
\[ F_{f} = \mu_k \cdot F_{N} \]
Here:

• \( \mu_k \) is the coefficient of kinetic friction, a value that depends on the types of materials in contact.
• \( F_{N} \) is the normal force, which is typically the weight of the object if the surface is level.

Kinetic friction is an essential concept because it determines how much force is needed to keep an object moving at a constant speed. If you know the coefficient of friction and the normal force, you can use this to calculate the frictional force and understand how much work is needed to overcome it.

Work-Energy Principle

In physics, work is done when a force causes an object to move. The Work-Energy Principle states that the work done by external forces on an object is equal to the change in its kinetic energy. For constant speed motion, the work done is calculated using:
\[ W = F \cdot d \cdot \cos(\Phi) \]
where:

• \( W \) is the work done.
• \( F \) is the force applied.
• \( d \) is the distance over which the force is applied.
• \( \Phi \) is the angle between the force and the direction of movement.

In cases where an object moves at a constant speed, even though its speed remains unchanged, work is still being done to counteract forces like friction. Understanding this principle helps explain why continuous input of energy (or work) is needed to maintain such motion, combating energy losses to friction.

Newton's Laws

Newton's Laws of Motion are foundational to understanding how objects behave under different forces:

• First Law: An object will remain at rest or move at a constant speed in a straight line unless acted upon by a net external force. This explains why a block moving at constant speed implies a balanced force scenario.
• Second Law: The acceleration of an object depends on two things - the net force acting on the object and the mass of the object. It is expressed by \( F = ma \), showing how force affects motion.
• Third Law: For every action, there is an equal and opposite reaction. This law explains interactions, such as why applying force on the block also results in friction against the motion.

In our exercise, these laws help us predict and understand how forces work together to maintain constant velocity despite the opposing frictional force. Newton's Laws provide the tools to analyze the balance of forces and motion, ensuring we can apply the right calculations accurately.