A 4.5kg box slides down a 4.1-m -high frictionless hill, starting from rest, across a 1.7-m -wide horizontal surface, then hits a horizontal spring with spring constant 490N/m . The other end of the spring is anchored against a wall. The ground under the spring is frictionless, but the 1.7-m-long horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.22.
A) What is the speed of the box just before reaching the rough surface?
B) What is the speed of the box just before hitting the spring?
C) How far is the spring compressed?
D) Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?
To solve this problem, we need to break it down into several steps and use principles of physics.
Step 1: Calculate the speed of the box just before reaching the rough surface (A).
Given:
- Mass of the box (m) = 4.5 kg
- Height of the frictionless hill (h) = 4.1 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Using conservation of energy, we can equate the potential energy at the top of the hill to the kinetic energy just before reaching the rough surface.
Potential Energy (PE) = mgh
Kinetic Energy (KE) = (1/2)mv^2
Since the box starts from rest, the initial kinetic energy is zero.
PE = KE
mgh = (1/2)mv^2
Canceling out the mass (m) from both sides of the equation:
gh = (1/2)v^2
Substituting the given values:
9.8 m/s^2 * 4.1 m = (1/2)v^2
Solving for v:
39.98 = (1/2)v^2
v^2 = 79.96
v = sqrt(79.96) = 8.94 m/s
Therefore, the speed of the box just before reaching the rough surface is 8.94 m/s.
Step 2: Calculate the speed of the box just before hitting the spring (B).
The box experiences friction on the rough surface. Let's calculate the friction force first.
The friction force (F_friction) can be expressed as:
F_friction = u * N
Where:
- u is the coefficient of kinetic friction = 0.22
- N is the normal force
The normal force (N) can be calculated using the weight of the box:
N = mg
N = 4.5 kg * 9.8 m/s^2 = 44.1 N
Substituting the values into the equation for friction force:
F_friction = 0.22 * 44.1 N = 9.702 N
The friction force acts opposite to the motion, so it acts in the opposite direction of the velocity vector.
Using the work-energy principle, we can equate the work done against friction to the change in kinetic energy.
Work done against friction = F_friction * d = (1/2)mv^2 - 0
Where:
- d is the distance across the rough surface
Solving for d:
9.702 N * 1.7 m = (1/2) * 4.5 kg * v^2
163.1496 J = 2.25 kg * v^2
Dividing both sides by 2.25 kg:
v^2 = 72.2773
v = sqrt(72.2773) = 8.50 m/s
Therefore, the speed of the box just before hitting the spring is 8.50 m/s.
Step 3: Calculate the distance the spring is compressed (C).
The potential energy stored in the spring is given by:
Potential Energy (PE_spring) = (1/2)kx^2
Where:
- k is the spring constant = 490 N/m
- x is the compression distance
Using the work-energy principle, we can equate the work done on the spring to the change in kinetic energy.
Work done on the spring = PE_spring - KE
(1/2)mv^2 = (1/2)kx^2
Since the box hits the spring from the right, the work done is positive and the kinetic energy is zero.
Substituting the values:
(1/2) * 4.5 kg * (8.50 m/s)^2 = (1/2) * 490 N/m * x^2
x^2 = (4.5 kg * (8.50 m/s)^2) / 490 N/m
x^2 = 131.67
x = sqrt(131.67) = 11.48 m
Therefore, the spring is compressed by 11.48 m.
Step 4: Calculate the number of complete trips across the rough surface before coming to rest (D).
The work done against friction in one complete trip across the rough surface is equal to the change in kinetic energy. After completing each trip, the box returns to rest.
Work done against friction = F_friction * d = (1/2)mv^2 - 0
Solving for d:
9.702 N * 1.7 m = (1/2) * 4.5 kg * (8.50 m/s)^2
d = (1/2) * [(4.5 kg * (8.50 m/s)^2) / 9.702 N]
d = 6.93 m
Therefore, the box will make approximately 1.7 m / 6.93 m ≈ 0.245 trips across the rough surface before coming to rest.
To solve this problem, we can use the principles of conservation of energy and the equations of motion.
A) To find the speed of the box just before reaching the rough surface, we need to calculate the potential energy the box has at the start and convert it into kinetic energy at the end of the hill.
First, let's calculate the potential energy at the start:
Initial potential energy (PEi) = mass (m) × gravitational acceleration (g) × height (h)
PEi = 4.5 kg × 9.8 m/s^2 × 4.1 m
Next, let's calculate the final kinetic energy just before reaching the rough surface:
Final kinetic energy (KEf) = 1/2 × mass (m) × speed^2
Since energy is conserved, the initial potential energy equals the final kinetic energy (there is no loss due to friction):
PEi = KEf
4.5 kg × 9.8 m/s^2 × 4.1 m = 1/2 × 4.5 kg × speed^2
Solving for speed:
speed^2 = (2 × 4.5 kg × 9.8 m/s^2 × 4.1 m) / 4.5 kg
speed = √(2 × 9.8 m/s^2 × 4.1 m)
B) To find the speed of the box just before hitting the spring, we can use the conservation of mechanical energy between the rough surface and the spring.
First, let's calculate the potential energy lost due to friction on the rough surface:
Potential energy lost (PE_loss) = kinetic friction coefficient (μ) × mass (m) × gravitational acceleration (g) × distance (d)
PE_loss = 0.22 × 4.5 kg × 9.8 m/s^2 × 1.7 m
Now, let's calculate the final kinetic energy just before hitting the spring:
Final kinetic energy (KEf) = initial kinetic energy (KEi) - PE_loss
Since there is no mention of any external force, we can assume the initial kinetic energy is equal to the kinetic energy just before reaching the rough surface (KEf from part A):
KEi = KEf = 1/2 × 4.5 kg × speed^2 - PE_loss
Solving for speed:
speed^2 = (2 × (1/2 × 4.5 kg × speed^2 - PE_loss)) / 4.5 kg
speed = √(2 × (1/2 × 4.5 kg × speed^2 - PE_loss))
C) To find how far the spring is compressed, we need to calculate the work done by the spring force on the box.
The work done by the spring force can be calculated using the formula:
Work (W) = 1/2 × spring constant (k) × (compression distance)^2
Since work done is equal to the change in kinetic energy:
Work (W) = 1/2 × mass (m) × (final velocity)^2 - 1/2 × mass (m) × (initial velocity)^2
Solving for the compression distance:
compression distance = √((1/2 × mass (m) × (final velocity)^2 - 1/2 × mass (m) × (initial velocity)^2) / (1/2 × spring constant (k)))
D) To determine the number of complete trips the box makes across the rough surface before coming to rest, we need to consider the loss of mechanical energy due to friction and the work done by the spring force.
Since the box will eventually come to rest, it means that the energy it loses due to friction is greater than the energy gained from the spring.
Therefore, you need to calculate the work done by the spring force as described in part C, and use it to find out how many complete trips the box can make before the energy loss due to friction equals the work done by the spring force.
By dividing the total work done on the box (mass × grav. accel. × distance) by the work done per trip, you can determine the number of complete trips the box makes before coming to rest.
Remember to use the value of compression distance calculated in part C to find the work done per trip.
Note: To get the exact values, you need to plug in the given values into the formulas and perform the calculations.