Math Problem Statement
Suppose the angles shown in the figure are 54 ∘ and 25 ∘. If the left-hand mass is 1.7 kg ,what should the right-hand mass be so that it accelerates downslope at 0.64 m/s2 ? Express your answer with the appropriate units.
Solution
To solve this problem, we'll break it down step by step, using Newton's laws and the given details to determine the unknown mass.
We are given the following information:
- Left-hand mass () = 1.7 kg
- Right-hand mass () = ?
- Acceleration () = 0.64 m/s²
- Angles: for the left-hand incline, and for the right-hand incline
Step 1: Write down the force components along the slopes for both masses
For each mass, the forces acting along the incline include gravity, tension in the string, and friction (if applicable, but none is mentioned here).
Left-hand mass () on the slope:
- Force due to gravity:
- Tension in the string:
- Net force:
Right-hand mass () on the slope:
- Force due to gravity:
- Tension in the string:
- Net force:
Step 2: Set up the system of equations
Since the tension is the same for both masses, we can set the two net forces equal but opposite in sign:
Step 3: Solve for the tension
From equation (1):
Step 4: Substitute into equation (2)
Substitute the expression for into equation (2):
Step 5: Solve for
Now, we can solve for :
Let me calculate the value of .The right-hand mass () should be approximately in order for the system to accelerate downslope at .
Would you like further details or clarification on any of the steps?
Here are some related questions you might find useful:
- What happens if the angles of the inclines change?
- How does the system behave if there is friction on the incline?
- Can the acceleration be negative, and how would that affect the system?
- How does changing one of the masses affect the acceleration?
- What if the pulley has mass or friction? How would the equations change?
Tip: Always double-check the angle conversion when solving problems involving slopes and inclines in trigonometric functions!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Newton's Laws of Motion
Inclined Plane
Trigonometry
Formulas
Net Force: F_net = m * a
Force due to gravity on incline: F_g = m * g * sin(θ)
Tension in the string and system of equations
Theorems
Newton's Second Law
Suitable Grade Level
Grades 11-12
Related Recommendation
Pulley System with Inclined Planes: Find Direction, Acceleration, and Tension
Physics Problem: Two Blocks on an Inclined Plane with Friction
Calculate Acceleration and Tension in an Inclined Plane System with Two Masses
Calculate Acceleration and Tension in Inclined Plane System with Friction
Calculate Acceleration and Velocity on an Inclined Plane with Initial Velocity