a horizontal force of magnitude 20.0 N is applied to a 3.05 kg book as the book slides a distance d = 0.87 m up a frictionless ramp at angle θ = 30°. The book begins with zero kinetic energy. What is its speed at the end of the displacement?
Well, let's get this straight. The book is sliding up a frictionless ramp, so it's basically on a joyride with no resistance. Lucky book! So, we have a force of 20.0 N being applied to this 3.05 kg book, making it move up the ramp with gusto.
Now, to find the speed of the book at the end of the displacement, we'll need to take into account the work done by the applied force. Work (W) is equal to force (F) times displacement (d) times the cosine of the angle (θ) between the force and displacement vectors. It's like a fancy way of calculating effort!
So, the work done on the book can be calculated as:
W = F * d * cos(θ)
Substituting the given values:
W = 20.0 N * 0.87 m * cos(30°)
Now, since the book starts with zero kinetic energy, all the work done is converted into the book's final kinetic energy. Energy can be found using the equation:
K.E. = 0.5 * m * v^2
Where m is the mass of the book and v is its final velocity (speed).
So, we have:
K.E. = W
0.5 * m * v^2 = 20.0 N * 0.87 m * cos(30°)
Now we just need to solve for v (the speed). Let's crunch some numbers, shall we?
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.
Given:
- Horizontal force applied to the book = 20.0 N
- Mass of the book = 3.05 kg
- Displacement along the ramp = 0.87 m
- Angle of the ramp = 30°
First, let's calculate the change in potential energy of the book.
Change in potential energy = m * g * h
Where:
m = mass of the book (3.05 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = vertical displacement of the book
Since the book is moving along a ramp, the vertical displacement h can be calculated using the formula:
h = d * sin(θ)
Where:
d = displacement along the ramp (0.87 m)
θ = angle of the ramp (30°)
h = 0.87 m * sin(30°)
h = 0.87 m * 0.5
h = 0.435 m
Now, plug in the values for m, g, and h:
Change in potential energy = 3.05 kg * 9.8 m/s^2 * 0.435 m
Next, calculate the work done on the book using the formula:
Work = Force * Distance
Work = 20.0 N * 0.87 m
Since the work done is equal to the change in potential energy, we can equate the two expressions:
20.0 N * 0.87 m = 3.05 kg * 9.8 m/s^2 * 0.435 m
Simplifying the equation:
17.4 Nm = 13.79 Nm
Now, we know the work done on the book, which is equal to its change in kinetic energy:
Change in kinetic energy = 13.79 Nm
Finally, we can calculate the final kinetic energy of the book using the formula:
Kinetic energy = 0.5 * m * v^2
Where:
m = mass of the book (3.05 kg)
v = final velocity of the book (unknown)
Since the book starts with zero kinetic energy, the change in kinetic energy is equal to the final kinetic energy. So, we can set the two expressions equal:
13.79 Nm = 0.5 * 3.05 kg * v^2
Simplifying the equation:
27.58 N = 1.525 kg * v^2
Divide both sides by 1.525 kg:
v^2 = 18.073
Take the square root of both sides to solve for v:
v = √(18.073)
v ≈ 4.25 m/s
Therefore, the speed of the book at the end of the displacement is approximately 4.25 m/s.
To find the speed of the book at the end of the displacement, we can use the principle of conservation of energy.
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 by the horizontal force is equal to the change in the book's kinetic energy.
1. Calculate the work done by the horizontal force:
The work done, W, is given by W = force * displacement * cos(theta), where theta is the angle between the force and the displacement. In this case, the force is 20.0 N and the displacement is along the ramp, so theta = 0°.
Therefore, we have W = 20.0 N * 0.87 m * cos(0°) = 20.0 N * 0.87 m * 1 = 17.4 J.
2. Calculate the change in kinetic energy:
The book starts with zero kinetic energy, so the change in kinetic energy is equal to the final kinetic energy.
We can use the formula for kinetic energy, K.E. = 0.5 * mass * velocity^2.
Let v be the final velocity of the book, so the final kinetic energy is 0.5 * 3.05 kg * v^2.
3. Equate the work done to the change in kinetic energy:
17.4 J = 0.5 * 3.05 kg * v^2.
4. Solve for v:
Divide both sides of the equation by 0.5 * 3.05 kg to isolate v^2:
v^2 = 17.4 J / (0.5 * 3.05 kg)
v^2 = 11.42 m^2/s^2
Take the square root of both sides to find the velocity:
v = √(11.42 m^2/s^2) ≈ 3.38 m/s
Therefore, the speed of the book at the end of the displacement is 3.38 m/s.
work done=20sin30*distance
PE gained=mgd*sin20
work done=PE gained+ KEgained
solvefor KE gained,then KE=1/2 m v^2 solve for v