a horizontal force of magnitude 18.0 N is applied to a 3.07 kg book as the book slides a distance d = 0.34 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?
To find the speed of the book at the end of the displacement, we need to apply the work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
In this case, the work done on the book is equal to the product of the applied force and the displacement: work = force * displacement * cos(angle)
Given:
Force = 18.0 N
Displacement = 0.34 m
Angle = 30°
Calculating the work done:
work = 18.0 N * 0.34 m * cos(30°)
work = 18.0 N * 0.34 m * 0.866
work = 5.6148 Joules
Since the book begins with zero kinetic energy, the entire work done on it will be converted into kinetic energy.
The kinetic energy is given by the equation: kinetic energy = 1/2 * mass * velocity^2
Plugging in the given values:
5.6148 Joules = 1/2 * 3.07 kg * velocity^2
Dividing both sides of the equation by (1/2 * 3.07 kg):
velocity^2 = 5.6148 Joules / (1/2 * 3.07 kg)
velocity^2 = 5.6148 Joules / 1.535 kg
Taking the square root of both sides of the equation to solve for velocity:
velocity = sqrt(5.6148 Joules / 1.535 kg)
velocity ≈ 2.43 m/s
Therefore, the speed of the book at the end of the displacement is approximately 2.43 m/s.
To find the speed of the book at the end of the displacement, we can use the principle of work and energy. The work done on an object is equal to the change in its kinetic energy.
First, let's find the work done on the book. The work is given by the formula:
Work = Force * Distance * Cosine(theta)
Here, the force applied is 18.0 N, the distance is 0.34 m, and theta is the angle of the ramp, which is 30°. Substituting the values into the formula:
Work = 18.0 N * 0.34 m * Cos(30°)
Now, let's find the change in kinetic energy. At the beginning, the book has zero kinetic energy, so the change in kinetic energy is equal to the kinetic energy at the end. Therefore, we have:
Change in Kinetic Energy = 0.5 * Mass * Velocity^2
Here, the mass of the book is 3.07 kg, and our goal is to find the velocity at the end.
Now, equating the work done with the change in kinetic energy, we have:
Work = Change in Kinetic Energy
18.0 N * 0.34 m * Cos(30°) = 0.5 * 3.07 kg * Velocity^2
Simplifying the equation:
6.12 N*m = 1.535 kg * Velocity^2
Now, let's solve for the velocity. Divide both sides of the equation by 1.535 kg:
Velocity^2 = (6.12 N*m)/(1.535 kg)
Velocity^2 = 3.99 m^2/s^2
Taking the square root of both sides:
Velocity = √(3.99 m^2/s^2)
Velocity ≈ 1.997 m/s
Therefore, the speed of the book at the end of the displacement is approximately 1.997 m/s.