. A 20-N crate starting at rest slides down a rough 5.0-m long ramp, inclined at 25 with the horizontal. 20 J of energy is lost to friction. What will be the speed of the crate at the bottom of the incline?
PE at the top is 20 * 5sin25°
KE at the bottom is PE - 20 = 1/2 mv^2
To find the speed of the crate at the bottom of the incline, we can use the principle of conservation of mechanical energy. The mechanical energy of the crate at the top of the incline is equal to the mechanical energy at the bottom, assuming no external forces act on the crate during its motion.
The total mechanical energy (E) is the sum of the potential energy (PE) and kinetic energy (KE) of the crate:
E = PE + KE
The potential energy of the crate at the top of the incline is given by:
PE = m * g * h
where m is the mass of the crate (20 N), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the incline.
The kinetic energy of the crate at the bottom of the incline is given by:
KE = 1/2 * m * v^2
where v is the velocity (speed) of the crate at the bottom.
Since the mechanical energy is conserved, we can equate the initial mechanical energy (at the top) to the final mechanical energy (at the bottom) and solve for v:
PE_initial + KE_initial = PE_final + KE_final
m * g * h + 0 = 0 + 1/2 * m * v^2
Simplifying this equation by canceling out like terms:
m * g * h = 1/2 * m * v^2
We can now solve for v:
v^2 = 2 * g * h
v = √(2 * g * h)
Now, we can substitute the given values into the equation to find the speed of the crate at the bottom of the incline:
v = √(2 * 9.8 m/s^2 * 5.0 m * sin(25°))
v = √(98 m^2/s^2 * 5.0 * 0.4226)
v = √209.356
v ≈ 14.47 m/s
Therefore, the speed of the crate at the bottom of the incline is approximately 14.47 m/s.
To find the speed of the crate at the bottom of the incline, we need to consider the energy conservation principle. Energy is lost due to friction, so we need to take that into account.
Here are the steps to find the speed of the crate at the bottom of the incline:
Step 1: Determine the initial potential energy of the crate.
The potential energy (PE) is given by the formula: PE = mgh, where m is the mass of the crate, g is the acceleration due to gravity, and h is the height of the ramp.
In this case, the height h can be determined using the formula: h = Lsinθ, where L is the length of the ramp and θ is the angle of the incline.
Given:
Mass (m) = 20 N (since weight = mass x gravity, and here gravity is roughly 10 m/s^2)
Length of the ramp (L) = 5.0 m
Angle of the ramp (θ) = 25 degrees
We need to convert the angle from degrees to radians:
θ (in radians) = θ (in degrees) * (π / 180)
θ (in radians) = 25° * (π / 180) ≈ 0.4363 radians
Now we can calculate the height of the ramp:
h = Lsinθ = 5.0 m * sin(0.4363) ≈ 2.632 m
Next, we can calculate the initial potential energy:
PE = mgh = 20 N * 2.632 m ≈ 52.64 J
Step 2: Determine the final kinetic energy of the crate.
The final kinetic energy (KE) can be calculated using the formula: KE = (1/2)mv^2, where v is the final velocity of the crate.
Since some energy is lost to friction, the final kinetic energy will be less than the initial potential energy. The amount of energy lost to friction is given as 20 J.
Final kinetic energy (KE) = Initial potential energy (PE) - Energy lost to friction
KE = PE - Energy lost to friction
KE = 52.64 J - 20 J = 32.64 J
Step 3: Solve for the final velocity.
We will equate the final kinetic energy to (1/2)mv^2 and solve for v.
32.64 J = (1/2) * 20 N * v^2
Dividing both sides by (1/2) * 20 N, and taking the square root, we get:
v^2 = (32.64 J) / ((1/2) * 20 N)
v^2 ≈ 3.264
Taking the square root, we find the final velocity of the crate at the bottom of the incline:
v ≈ √3.264 ≈ 1.806 m/s
Therefore, the speed of the crate at the bottom of the incline is approximately 1.806 m/s.