Question

A 0.50 -kg physics textbook is hanging from two massless wires of equal length attached to a ceiling. The tension on each wire is measured as $15.4\mathrm{N}$. What is the angle of the wires with the horizontal?

Short answer

Answer: The angle the wires make with the horizontal is approximately 18.67°.

Step-by-step solution

Write down the given variables and determine what needs to be found

We are given: - Mass of the textbook (m) = 0.50 kg - Tension on each wire (T) = 15.4 N We need to find the angle (θ) the wires make with the horizontal.

Calculate the gravitational force acting on the textbook

First, let's find the gravitational force acting on the textbook. The gravitational force (Fg) can be calculated using the equation: Fg = mg where 'm' is the mass and 'g' is the gravitational acceleration (approximately $9.81\mathrm{m}/{\mathrm{s}}^2$). Fg = (0.50 kg) · (9.81 m/s^2) = 4.905 N

Apply Newton's second law and geometry principles

Since the textbook hangs symmetrically, we can treat half of the scenario as a right triangle. Using the properties of right triangles, we can relate the horizontal and vertical components of the tension force: - Vertical component of tension (T_y): $T_y=T\sin (\theta )$ - Horizontal component of tension (T_x): $T_x=T\cos (\theta )$ Applying Newton's second law in the vertical direction: ΣF_y = 0 (The system is at rest) So, $T\sin (\theta )-Fg=0$

Solve for the angle θ

Now, we will solve for the angle θ. Rearrange the equation from step 3: $T\sin (\theta )=Fg$ Divide both sides by T: $\sin (\theta )={\displaystyle \frac{Fg}{T}}$ Now, calculate the sine of θ: $\sin (\theta )={\displaystyle \frac{4.905\mathrm{N}}{15.4\mathrm{N}}}=0.3182$ To find the angle θ, take the inverse sine: θ = ${\sin }^{-1}(0.3182)$ θ ≈ 18.67° So, the angle the wires make with the horizontal is approximately 18.67°.

Key concepts

Newton's Second Law

Newton's Second Law is a critical principle in physics, defining how the motion of an object is affected by forces. This law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, expressed as $F=ma$. In scenarios where systems are at rest or moving at constant velocity, this principle implies that the sum of the forces is zero. This is vital when analyzing forces like tension, as it allows us to equate them with gravitational forces or any other external forces acting on the object. By ensuring that the net force is zero, we can solve for unknowns using geometry or trigonometry, as done in the provided exercise.

Tension Force

The tension force in a wire or rope is the force that is transmitted through it when it is pulled tight by forces acting from opposite ends. It's a pulling force directed along the length of the wire, and it transmits external forces to an object attached to the wire. In our exercise, the wires holding the textbook must bear the gravitational force of the textbook by providing an upward tension force. These tension forces have components both in vertical and horizontal directions. The vertical component counteracts gravity, while the horizontal components balance each other out due to the symmetrical setup.

Gravitational Force

Gravitational force is the force of attraction between two masses. For objects near the earth's surface, it is the force exerted by Earth's gravity on an object, calculated using $F_g=mg$, where $m$ is the mass and $g$ is the gravitational acceleration, approximately $9.81{\text{m/s}}^2$. In our problem, the textbook's weight or gravitational force is found by multiplying its mass (0.50 kg) by the gravitational acceleration, resulting in approximately 4.905 N. This force pulls the book downwards and must be balanced by an upward force, such as the vertical components of the tension forces, to keep the book hanging stationary.

Right Triangle Geometry

A right triangle is a simple geometric figure with one angle equal to ${90}^{\circ }$. It allows us to use trigonometric ratios to calculate unknown sides or angles. In this exercise, we can conceptualize half of the hanging system as forming a right triangle. The wires create a triangle where the horizontal distance from the top to the centerline serves as one leg, and the vertical distance as another. The tension in the wires acts as the hypotenuse. Using this setup and the relationship between the components of the tension (horizontal and vertical), we can employ trigonometric functions to solve for the angle the wires make with the horizontal.

Trigonometry in Physics

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. It is crucial in physics for resolving vector quantities, like forces, into their components. In the textbook problem, trigonometric functions relate the tension force to its components. The vertical component of tension is found using $T_y=T\sin (\theta )$, while the horizontal is $T_x=T\cos (\theta )$. These functions are particularly useful when balancing forces, as they allow us to express components in terms of familiar mathematical functions, making it easier to solve for unknowns like angles or tensions in physics problems.