Math Problem Statement
Problem 5: (7% of Assignment Value) A horizontal force, F1 = 55 N, and a force, F2 = 10.3 N acting at an angle of θ to the horizontal, are applied to a block of mass m = 3.7 kg. The coefficient of kinetic friction between the block and the surface is μk = 0.2. The block is moving to the right.
Randomized Variables F1 = 55 N F2 = 10.3 N m = 3.7 kg
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Part (a) ✔ Solve numerically for the magnitude of the normal force, FN in Newtons, that acts on the block if θ = 30°. FN = 41.45 ✔ Correct!
Part (b) Solve numerically for the magnitude of acceleration of the block, a in m/s2, if θ = 30°. a =
Solution
To solve for the magnitude of acceleration of the block, we need to analyze the forces acting on it and apply Newton's Second Law.
Step 1: Analyze the Forces
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Force : This is a horizontal force of 55 N.
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Force : This force has a magnitude of 10.3 N and is applied at an angle to the horizontal. We can break this force into its horizontal and vertical components:
- Horizontal component of :
- Vertical component of :
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Frictional Force: The frictional force opposes the motion and can be calculated using the coefficient of kinetic friction and the normal force (which was previously calculated as 41.45 N):
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Normal Force : The normal force was already determined as 41.45 N in part (a).
Step 2: Newton’s Second Law
To find the acceleration, we apply Newton’s Second Law in the horizontal direction. The forces in the horizontal direction are:
- The horizontal component of ,
- The frictional force,
The net force in the horizontal direction is:
Using Newton’s Second Law: Thus, the equation for acceleration is:
Step 3: Calculation
Let’s calculate each of the components:
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Horizontal component of :
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Frictional force:
Now, we can find the net force:
Finally, using Newton’s Second Law to find the acceleration:
Final Answer:
The magnitude of the acceleration is approximately:
Would you like further details or clarifications on this solution? Here are a few related questions you could explore:
- How would the acceleration change if the angle were increased to 45°?
- What would happen to the normal force if the coefficient of friction were increased to 0.3?
- How can we calculate the work done by the frictional force on the block?
- What would the acceleration be if the block had a different mass, say 5 kg?
- How can the net force be affected if the direction of is reversed?
Tip: When calculating frictional forces, always ensure you're using the correct normal force, as it directly influences the frictional interaction between surfaces.
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Math Problem Analysis
Mathematical Concepts
Newton's Second Law
Friction
Vector Decomposition
Kinematics
Formulas
F_{2x} = F_2 * cos(θ)
F_{2y} = F_2 * sin(θ)
F_{friction} = μk * F_N
F_{net} = F_1 + F_{2x} - F_{friction}
a = F_{net} / m
Theorems
Newton's Second Law
Suitable Grade Level
Grades 11-12
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