A student, crazed by the first physics Midterm, uses a pushing force of
80 N angled at 70° to push a 5.0 kg book across the frictionless ceiling
of his room.
a. What is the block's acceleration?
b. What is the force of the ceiling on the book?
Is the force applied at 70 degrees to the ceiling or the surface normal?
I assume it is 70 degrees to the ceiling, otherwise the brick would fall. The normal force on the ceiling is then 80sin70-5g = 26.1 N
(That is the answer to part (b))
The component along the ceiling is then
80 cos70 = 27.4 N
Acceleration = 27.4 N/(mass)= 5.47 m/s^2
To find the answers to these questions, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Here's how we can proceed:
a. What is the block's acceleration?
Step 1: Resolve the pushing force into its vertical and horizontal components.
- The vertical component can be found by multiplying the pushing force by the sine of the angle: F_vertical = F * sin(θ).
- The horizontal component can be found by multiplying the pushing force by the cosine of the angle: F_horizontal = F * cos(θ).
- In this case, the vertical component would be: F_vertical = 80 N * sin(70°).
- The horizontal component would be: F_horizontal = 80 N * cos(70°).
Step 2: Identify the forces acting in the vertical and horizontal directions.
- The only force acting vertically is the weight of the book, which is equal to its mass times the acceleration due to gravity: F_weight = m * g.
- The only horizontal force is the horizontal component of the pushing force.
Step 3: Set up the equations of motion.
- In the vertical direction, the weight of the book equals the vertical component of the pushing force: F_vertical = F_weight.
- In the horizontal direction, the horizontal component of the pushing force equals the product of the book's mass and its acceleration: F_horizontal = m * a.
Step 4: Solve for the acceleration.
- Substitute the known values into the equation from step 3: 80 N * sin(70°) = 5.0 kg * a.
- Rearrange the equation to solve for a: a = (80 N * sin(70°)) / 5.0 kg.
- Evaluate the expression to find the acceleration in meters per second squared (m/s²).
b. What is the force of the ceiling on the book?
Step 1: From step 2, we determined that the only vertical force acting on the book is its weight.
- The force of the ceiling on the book will be equal in magnitude and opposite in direction to the weight of the book.
- Therefore, the force of the ceiling on the book is equal to the weight of the book (F_weight).
Step 2: Calculate the weight of the book.
- The weight of an object is given by the equation: F_weight = m * g.
- Substitute the known values into the equation: F_weight = 5.0 kg * g.
- Evaluate the expression to find the force of the ceiling on the book in newtons (N).
By following these steps, we can determine the block's acceleration and the force of the ceiling on the book.