A horizontal force of magnitude 22 N is applied to a 3.0 kg book as the book slides a distance d = 0.5 m up a frictionless ramp at angle è = 30°. The book begins with zero kinetic energy. What is its speed at the end of the displacement?
No friction so any work done on it = change in PE + change in KE
Assume both PE and KE are zero at start.
How far up did it go?
h = .5 sin 30 = .25 meters up
How much potential energy did it gain?
PE = m g h = 3.0 * 9.8 * .25 = 7.35 Joules
How much work was done on it
Force in direction of motion = 22 cos 30 = 19.1 N
work done = 19.1 * .5 = 9.53 joules
so
9.53 = KE + 7.35
so
(1/2) m v^2 = 2.18 Joules
I think you can get v from there
To find the speed of the book at the end of the displacement, we need to apply the principles of work and energy.
First, let's find the work done on the book by the applied force. The work is given by the formula:
Work = Force * Displacement * cos(θ)
Where:
- Work is the amount of energy transferred to the book
- Force is the magnitude of the applied force (22 N)
- Displacement is the distance the book is moved (0.5 m)
- θ is the angle between the force and displacement (30°)
Plugging in the values, we have:
Work = 22 N * 0.5 m * cos(30°)
Next, we need to account for the change in potential energy as the book moves up the ramp. The change in potential energy is given by the formula:
Change in Potential Energy = mass * acceleration due to gravity * height
Where:
- mass is the mass of the book (3.0 kg)
- acceleration due to gravity is approximately 9.8 m/s^2
- height is the vertical distance the book moves
Since the book is moving along a ramp at an angle of 30°, the height can be found using the formula:
height = displacement * sin(θ)
Plugging in the value of the displacement and angle, we have:
height = 0.5 m * sin(30°)
Now, let's calculate the change in potential energy:
Change in Potential Energy = 3.0 kg * 9.8 m/s^2 * (0.5 m * sin(30°))
Finally, using the principle of conservation of energy, the work done on the book will be equal to the change in potential energy plus the change in kinetic energy:
Work = Change in Potential Energy + Change in Kinetic Energy
Since the book starts with zero kinetic energy, we can simplify the equation to:
Change in Kinetic Energy = Work - Change in Potential Energy
Now, we can rearrange the equation to solve for the speed at the end of displacement:
Speed = sqrt((2 * (Work - Change in Potential Energy)) / mass)
Plugging in the values for Work, Change in Potential Energy, and mass, we can calculate the speed.