Question

A child pulls a friend in a little red wagon. If the child pulls with a force of \(16 \mathrm{~N}\) for \(12 \mathrm{~m}\) and the handle of the wagon is inclined at an angle of \(25^{\circ}\) above the horizontal, how much work does the child do on the wagon?

Short answer

The child does 174.82 Joules of work.

Step-by-step solution

Understand the Concept of Work

Work is done when a force causes an object to move in the direction of the force. The amount of work is calculated using the formula: \[W = F \cdot d \cdot \cos(\theta)\]where \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle between the force and the direction of motion.

Identify the Given Values

From the problem, we know the following values: - Force (\( F \)) = 16 N- Distance (\( d \)) = 12 m- Angle (\( \theta \)) = 25 degrees.

Convert Angle to Radians if Necessary

In this problem, we will use the angle in degrees directly because most trigonometric functions can use degrees. However, ensure your calculator is in the correct mode for degrees.

Calculate the Work Done Using the Formula

Substitute the given values into the work formula:\[W = 16 \times 12 \times \cos(25^{\circ})\]Calculate \( \cos(25^{\circ}) \) using a calculator.

Solve for Work

Compute \( \cos(25^{\circ}) \approx 0.9063 \). Now substitute this value back into the equation:\[W = 16 \times 12 \times 0.9063 \approx 174.82 \, \text{Joules}\]Thus, the work done by the child is approximately 174.82 Joules.

Key concepts

force and motion

Force and motion are two core concepts in physics that describe how objects move. When a force is applied to an object, it can cause the object to start moving, stop moving, or change its motion. The relationship between force and motion is governed by Newton's laws of motion.

• Newton's First Law: An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force.
• Newton's Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This can be expressed with the formula: \( F = m imes a \), where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.
• Newton's Third Law: For every action, there is an equal and opposite reaction.

In the exercise, the child applies a force to the wagon, which causes it to move. The force of 16 N is what gets the wagon in motion for the 12 m distance.

angle of inclination

The angle of inclination refers to the angle at which a force is applied concerning the horizontal direction. In the provided exercise, the angle of inclination of the wagon's handle is 25 degrees above the horizontal.

Angles affect how effectively a force contributes to moving an object forward. Only a component of the force is used to move the object in the direction of motion when there is an angle present. This is why understanding the angle of inclination is essential to accurately calculate the work done on an object.

For example, if the force is applied horizontally, all of the force contributes to moving the object. However, when a force is applied at an angle, only the horizontal component (calculated as \( F imes \cos(\theta) \)) helps in doing work.

work formula

The work formula is a key concept in physics to calculate the work done by a force. It is given by:

\[W = F imes d imes \cos(\theta)\]

Where:

• \( W \) denotes the work done measured in Joules.
• \( F \) is the force applied in Newtons.
• \( d \) is the distance over which the force is applied, measured in meters.
• \( \theta \) is the angle between the force and the direction of motion.

Using the exercise example, substituting the values into the formula involves plugging 16 N for the force, 12 m for distance, and using \( \cos(25^\circ) \) in the calculation. The calculated cosine value helps in determining how much of the force effectively works to move the wagon. This results in approximately 174.82 Joules of work done.

trigonometric functions

Trigonometric functions are mathematical tools that relate the angles and sides of triangles. In physics, they help in breaking down a force vector into its components. The most commonly used trigonometric functions are sine, cosine, and tangent.

• Cosine function (\(\cos\)): Used in work calculations to find the horizontal component of the applied force. In the formula \(W = F imes d imes \cos(\theta)\), it helps determine the effective part of the force that does the work. For an angle of 25 degrees, \(\cos(25^\circ)\) is approximately 0.9063.
• Sine function (\(\sin\)): Gives the vertical component of a force, which is useful in other types of calculations involving inclined planes or when finding frictional forces.
• Tangent function (\(\tan\)): Not directly used in this exercise, but it relates the sine and cosine of an angle and can be used to solve different types of problems involving slopes and angles.

Understanding these functions is crucial in physics to analyze forces acting in various directions and to solve real-world problems involving angles.