Question

In a plate glass factory, sheets of glass move along a conveyor belt at a speed of \(15.0 \mathrm{cm} / \mathrm{s}\). An automatic cutting tool descends at preset intervals to cut the glass to size. since the assembly belt must keep moving at constant speed, the cutter is set to cut at an angle to compensate for the motion of the glass. If the glass is \(72.0 \mathrm{cm}\) wide and the cutter moves across the width at a speed of \(24.0 \mathrm{cm} / \mathrm{s},\) at what angle should the cutter be set?

Short answer

The cutter should be set at an angle of approximately 51.3°.

Step-by-step solution

Understand the Problem

We need to find the angle at which the cutting tool should be set to cut across a moving sheet of glass that is 72.0 cm wide. The glass moves at a speed of 15.0 cm/s along the conveyor belt, while the cutter moves across the width at 24.0 cm/s.

Set Up the Equation

The cutter needs to move diagonally to cut across a moving sheet. The angle of the cutter, \(\theta\), can be found using trigonometry. To maintain a straight cut across the moving glass, the horizontal component of the cutter's velocity must match the speed of the conveyor belt, \(v = 15.0\, \text{cm/s}\). The cutter's speed across the width is \(v_c = 24.0\, \text{cm/s}\). We can write:\[ \cos(\theta) = \frac{v}{v_c} = \frac{15.0}{24.0} \]

Calculate the Angle

Substitute the given values into the cosine equation:\[ \cos(\theta) = \frac{15.0}{24.0} = 0.625 \]Now, find \(\theta\) by taking the arccosine:\[ \theta = \cos^{-1}(0.625) \approx 51.3^\circ \]

Verify the Result

Recheck the setup of the equation and calculations to ensure that all values were correctly used. The cutter is set to compensate for 15.0 cm/s horizontally and move across a 72.0 cm wide glass at 24.0 cm/s vertically along the conveyor. The math and interpretation align with the problem requirements.

Key concepts

Trigonometry in Physics

Trigonometry is a branch of mathematics dealing with angles and sides of triangles. In the realm of physics, it is crucial for solving problems involving angles and motion. When discussing conveyor belts, the movement often involves different directions, requiring trigonometry to find solutions.
In this specific problem, trigonometry helps us determine the angle at which the cutter should be oriented to accurately cut moving glass. We use the trigonometric function, cosine, which relates the adjacent side to the hypotenuse in a right triangle, to solve this. Here, the horizontal component of the velocity is the adjacent side, and the cutter's speed (across the width) is the hypotenuse.
By setting up the equation and solving for the angle using the cosine inverse, we employ the foundational trigonometric principle to find the angle needed for a precise cut. Without trigonometry, problems involving angles and directional motions would be challenging to solve.

Conveyor Belt Mechanics

A conveyor belt is a vital part of many manufacturing processes, helping to move materials efficiently. In our problem, the conveyor belt's role is to continuously move glass sheets at a constant speed of 15.0 cm/s. This ongoing motion requires tools like cutters to be precisely synchronized.
The cutter must cut the glass without slowing down the belt, which means it must move at an angle to adjust for both its own speed and the belt's horizontal speed. This requires a deep understanding of how both components work together. If the cutter were set straight across, while the glass continues to move horizontally, the cut line would be slanted.
In situations like these, mechanics ensures that devices and tools align with the required movements. Conveyor belt mechanics highlight the critical need for proper adjustments in tool settings to maintain accuracy and efficiency when working with continuous motion.

Velocity Components

Understanding velocity components is essential when dealing with motion in multiple directions. Velocity is a vector, meaning it has both magnitude and direction. In physics, especially with problems like our conveyor belt scenario, we break it into components to simplify calculations.
In this problem, the velocity of the cutter is broken into two components: horizontal and vertical. The horizontal component matches the speed of the conveyor belt (15.0 cm/s), while the vertical component reflects the cutter moving across the width (24.0 cm/s). By investigating these separately, we can apply trigonometric functions effectively.
The separation into components allows for clear analysis. With the horizontal velocity given, we use trigonometry, notably \( \cos(\theta) \), to find necessary angles that ensure the cutting tool works seamlessly with the motion of the glass. Mastering velocity components is key to resolving complex motion scenarios in physics.