Math Problem Statement
Three objects are connected on a table. The coefficient of static friction between m3 and the table is 0.350. The objects have a masses m1=4 kg, m2=1 kg, and m3=2 kg. Determine the acceleration of the objects. Round to the nearest tenths, units!
Solution
Let's break down the problem step by step. We have three masses connected on a table. The goal is to find the acceleration of the system. We'll assume the setup is something like this:
- Masses , , and are connected by strings.
- and are hanging off the edges of the table (each on one side of the table), while is on the surface of the table.
- There's a coefficient of static friction between and the table, given as .
We'll follow these steps:
Step 1: Identify forces acting on the system.
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The weight of each mass is , where .
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For mass and , they will experience gravitational forces pulling them down:
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For mass , it experiences two forces: tension in the strings and friction from the table surface. The friction force is given by , where is the normal force.
Step 2: Write the equations of motion.
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For mass (moving downward): where is the tension in the string attached to , and is the acceleration of the system.
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For mass (moving downward): where is the tension in the string attached to .
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For mass (on the table): where .
Step 3: Solve for acceleration .
First, let's calculate the friction force on :
Now we can set up the system of equations:
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Substitute and into this equation: Simplify:
Final Answer:
The acceleration of the system is approximately 4.5 m/s².
Would you like more details on the steps or equations? Here are five follow-up questions to consider:
- How does the coefficient of friction affect the overall acceleration of the system?
- What happens to the acceleration if the mass is increased?
- How would the setup change if and were on different surfaces with different friction coefficients?
- Can you calculate the tensions in the strings after finding the acceleration?
- How would the results differ if there was no friction between and the table?
Tip: Always check the direction of friction. It opposes the motion of the object on the surface!
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Math Problem Analysis
Mathematical Concepts
Newton's Second Law
Forces
Friction
Tension in Strings
Formulas
F = m * a
F_friction = μ * N
N = m * g
Tension equations for m1, m2, and m3
Theorems
Newton's Second Law of Motion
Suitable Grade Level
Grades 10-12
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