A freight company uses a compressed spring to shoot 2.00 kg packages up a 1.00-m -high frictionless ramp into a truck, as the figure shows. The spring constant is 481 N/m and the spring is compressed 30.0 cm .
A. What is the speed of the package when it reaches the truck?
B. A careless worker spills his soda on the ramp. This creates a 50.0--long sticky spot with a coefficient of kinetic friction 0.300. Will the next package make it into the truck?
A. To find the speed of the package when it reaches the truck, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of an object remains constant if no external forces are acting on it.
The total mechanical energy of the package can be calculated as the sum of its potential energy (due to its height above the ground) and its kinetic energy.
Potential Energy (PE) = m * g * h
where m is the mass of the package (2.00 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ramp (1.00 m).
PE = 2.00 kg * 9.8 m/s^2 * 1.00 m = 19.6 J
The compressed spring stores potential energy, which is then converted into the kinetic energy of the package as it gets launched up the ramp.
The potential energy stored in the spring can be calculated using the formula:
Potential Energy (PE) = (1/2) * k * x^2
where k is the spring constant (481 N/m) and x is the compression distance of the spring (30.0 cm = 0.30 m).
PE = (1/2) * 481 N/m * (0.30 m)^2 = 21.645 J
Since the total mechanical energy is conserved, the potential energy at the start (stored in the compressed spring) is equal to the total energy at the end (kinetic energy of the package).
Kinetic Energy (KE) = Total Mechanical Energy - Potential Energy
KE = 21.645 J - 19.6 J = 2.045 J
To find the speed (v) of the package when it reaches the truck, we use the formula for kinetic energy:
KE = (1/2) * m * v^2
2.045 J = (1/2) * 2.00 kg * v^2
v^2 = 2.045 J * 2 / (2.00 kg) = 2.045 m^2/s^2
v = √(2.045 m^2/s^2) ≈ 1.43 m/s
Therefore, the speed of the package when it reaches the truck is approximately 1.43 m/s.
B. To determine whether the next package will make it into the truck, we need to account for the effect of the sticky spot with a coefficient of kinetic friction of 0.300.
The work performed by the force of friction is given by:
Work (W) = Force of friction (f) * distance (d)
The force of friction can be calculated using the formula:
f = μ * N
where μ is the coefficient of kinetic friction (0.300) and N is the normal force.
The normal force is equal to the weight of the package, which can be calculated as:
Weight (W) = m * g
where m is the mass of the package (2.00 kg) and g is the acceleration due to gravity (9.8 m/s^2).
W = 2.00 kg * 9.8 m/s^2 = 19.6 N
Therefore, the normal force is 19.6 N.
Now we can calculate the force of friction:
f = 0.300 * 19.6 N = 5.88 N
The distance over which the force of friction acts is the length of the sticky spot, which is given as 50.0 cm = 0.50 m.
Now we can calculate the work performed by the force of friction:
W = f * d = 5.88 N * 0.50 m = 2.94 J
The work done by the force of friction removes mechanical energy from the system.
Since the work done by the force of friction is positive, it acts in the opposite direction of the motion. As a result, the next package will not make it into the truck, as some of the mechanical energy is lost due to the work done by friction.
To find the speed of the package when it reaches the truck (Question A), we can use the principle of conservation of energy. We will consider the potential energy of the spring when compressed and the kinetic energy of the package when it reaches the truck.
A. First, let's find the potential energy stored in the compressed spring. The potential energy of a spring can be calculated using the formula:
PE = (1/2)kx^2
Where PE is the potential energy, k is the spring constant, and x is the distance the spring is compressed.
Given:
k = 481 N/m
x = 30.0 cm = 0.3 m
PE = (1/2)(481 N/m)(0.3 m)^2 = 21.735 J
The potential energy stored in the spring is 21.735 Joules.
Next, let's convert this potential energy into kinetic energy when the package reaches the truck. Since the ramp is frictionless, the only form of energy the package will have is kinetic energy.
KE = PE
KE = 21.735 J
The kinetic energy of the package will also depend on its mass and velocity. We can equate the kinetic energy to the kinetic energy formula:
KE = (1/2)mv^2
Where m is the mass of the package and v is its velocity.
Given:
m = 2.00 kg
Substituting the values we have:
(1/2)(2.00 kg)v^2 = 21.735 J
Simplifying the equation:
v^2 = (2 * 21.735 J) / (2.00 kg)
v^2 = 21.735 J / 2.00 kg
v^2 = 10.8675 m^2/s^2
Taking the square root of both sides to find the velocity:
v = √(10.8675 m^2/s^2)
v ≈ 3.295 m/s
The speed of the package when it reaches the truck is approximately 3.295 m/s.
B. To determine whether the next package will make it into the truck (Question B), we need to consider the presence of the sticky spot and the coefficient of kinetic friction.
The force of friction can be calculated using the formula:
F_friction = μ * N
Where F_friction is the force of friction, μ is the coefficient of kinetic friction, and N is the normal force.
Given:
μ = 0.300
N = mg
Let's assume that g = 9.8 m/s^2 (acceleration due to gravity).
Since the package is on an inclined ramp, the normal force can be found using:
N = mg * cosθ
Where θ is the angle of the ramp. Since the ramp is not specified, for simplicity, we will assume it is a 30 degrees incline.
θ = 30 degrees
N = (2.00 kg)(9.8 m/s^2) * cos(30 degrees)
N ≈ 31.92 N
Substituting these values back into the force of friction equation:
F_friction = 0.300 * 31.92 N
F_friction ≈ 9.576 N
Now, we can calculate the force applied by the compressed spring:
F_spring = k * x
Given:
k = 481 N/m
x = 30.0 cm = 0.3 m
F_spring = 481 N/m * 0.3 m
F_spring = 144.3 N
Comparing the force of friction and the force of the spring:
F_spring > F_friction
144.3 N > 9.576 N
Since the force applied by the compressed spring is greater than the force of friction, the next package will make it into the truck, even with the sticky spot on the ramp.