A block slides down a frictionless incline of angle 30 degrees. If the incline has a coefficient of kinetic friction of 0.2, what is the acceleration of the block?
Practice Questions
1 question
Q1
A block slides down a frictionless incline of angle 30 degrees. If the incline has a coefficient of kinetic friction of 0.2, what is the acceleration of the block?
4.9 m/s²
3.9 m/s²
2.9 m/s²
1.9 m/s²
Net force = mg sin(30) - μmg cos(30). Acceleration a = (mg sin(30) - μmg cos(30))/m = g(sin(30) - μ cos(30)). Substituting g = 10 m/s² gives a = 10(0.5 - 0.2 * √3/2) = 4.9 m/s².
Questions & Step-by-step Solutions
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Q
Q: A block slides down a frictionless incline of angle 30 degrees. If the incline has a coefficient of kinetic friction of 0.2, what is the acceleration of the block?
Solution: Net force = mg sin(30) - μmg cos(30). Acceleration a = (mg sin(30) - μmg cos(30))/m = g(sin(30) - μ cos(30)). Substituting g = 10 m/s² gives a = 10(0.5 - 0.2 * √3/2) = 4.9 m/s².
Steps: 10
Step 1: Identify the forces acting on the block. The block experiences gravitational force down the incline and frictional force opposing the motion.
Step 2: Calculate the gravitational force component acting down the incline. This is given by mg sin(30 degrees). Since sin(30) = 0.5, this becomes mg * 0.5.
Step 3: Calculate the normal force acting on the block. The normal force is given by mg cos(30 degrees). Since cos(30) = √3/2, this becomes mg * (√3/2).
Step 4: Calculate the frictional force. The frictional force is given by the coefficient of kinetic friction (μ) times the normal force. This is μ * mg cos(30 degrees) = 0.2 * mg * (√3/2).
Step 5: Write the net force equation. The net force acting on the block is the gravitational force down the incline minus the frictional force: Net force = mg sin(30) - μmg cos(30).
Step 6: Substitute the values into the net force equation. This gives us Net force = mg * 0.5 - 0.2 * mg * (√3/2).
Step 7: Factor out mg from the equation. This simplifies to Net force = mg (0.5 - 0.2 * (√3/2)).
Step 8: Use Newton's second law (F = ma) to find acceleration. Since F = ma, we can write a = Net force / m = g (0.5 - 0.2 * (√3/2)).
Step 9: Substitute g = 10 m/s² into the equation for acceleration. This gives a = 10 * (0.5 - 0.2 * (√3/2)).
Step 10: Calculate the final value for acceleration. After performing the calculations, we find a = 4.9 m/s².
