The coefficient of static friction is 0.604 between two blocks. The coefficient of kinetic friction between the lower block and the floor is 0.115. Force F causes both blocks to cross a distance of 3.93 m, starting from rest. What is the least amount of time in which the motion can be completed without the top block sliding on the lower block, if the mass of the lower block is 1.03 kg and the mass of the upper block is 3.04 kg?
To find the least amount of time in which the motion can be completed without the top block sliding on the lower block, we need to consider a few concepts from physics.
Let's break down the problem step by step:
Step 1: Calculate the maximum force of static friction (F_max):
The maximum force of static friction can be calculated using the formula:
F_max = μ_s * N
Where:
- μ_s is the coefficient of static friction (given as 0.604)
- N is the normal force acting on the top block
In this case, the normal force acting on the top block is equal to the weight of the block, which can be calculated as:
N = m_top * g
Where:
- m_top is the mass of the top block (given as 3.04 kg)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
So, we can calculate the maximum force of static friction (F_max) as:
F_max = 0.604 * (3.04 * 9.8)
Step 2: Calculate the required force (F_req) to move both blocks:
To prevent the top block from sliding on the lower block, the force applied (F_req) must be less than or equal to the maximum force of static friction (F_max).
So, F_req ≤ F_max
Step 3: Calculate the work done on the system:
The work done on the system is given by the formula:
Work = Force * Distance
In this case, the force applied is F_req (as determined in Step 2), and the distance traveled is 3.93 m (given in the question).
Step 4: Calculate the time required to complete the motion:
The time required to complete the motion can be calculated using the equation:
Work = Force * Distance = (1/2) * m * v^2
Where:
- m is the mass of the system (total mass of both blocks)
- v is the final velocity of the system (which is equal to the velocity of the blocks at the end of the motion)
We can solve this equation for time (t):
t = Distance / Velocity
To find the final velocity (v), we need to use the concept of kinetic friction:
The force of kinetic friction (F_k) can be calculated using the formula:
F_k = μ_k * N
Where:
- μ_k is the coefficient of kinetic friction (given as 0.115)
- N is the normal force acting on the lower block
In this case, the normal force acting on the lower block is equal to the weight of the block, which can be calculated as:
N = m_lower * g
Where:
- m_lower is the mass of the lower block (given as 1.03 kg)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
So, we can calculate the force of kinetic friction (F_k) as:
F_k = 0.115 * (1.03 * 9.8)
Finally, we can use the concept of net force to find the force required to move the system (F_req):
F_req = F_k + m_total * a
Where:
- m_total is the total mass of the system (sum of the masses of the top and lower block)
- a is the acceleration of the system (which can be determined using the equation: a = (v_final - v_initial) / t)
Since the system starts from rest (v_initial = 0) and travels a distance of 3.93 m, we can solve for the acceleration (a):
a = (0 - v_initial) / t = -v_final / t
Now, we can substitute the values and solve for time (t).
To find the least amount of time in which the motion can be completed without the top block sliding on the lower block, we need to analyze the forces acting on the system.
1. Determine the normal force on the lower block:
The normal force is equal to the weight of the block, which can be calculated using the formula:
normal force = mass x gravity, where gravity is approximately 9.8 m/s^2.
For the lower block:
normal force on lower block = mass of lower block x gravity
normal force on lower block = 1.03 kg x 9.8 m/s^2
2. Calculate the maximum static friction force:
The maximum static friction force can be calculated using the formula:
maximum static friction force = coefficient of static friction x normal force
maximum static friction force = 0.604 x normal force on lower block
3. Determine the minimum force required to keep the blocks from sliding:
The minimum force required to keep the blocks from sliding is equal to the maximum static friction force.
minimum force = maximum static friction force
minimum force = 0.604 x normal force on lower block
4. Calculate the acceleration of the system:
Using Newton's second law (F = ma), we can solve for acceleration.
minimum force = (mass of lower block + mass of upper block) x acceleration
acceleration = minimum force / (mass of lower block + mass of upper block)
5. Calculate the time required to travel the distance:
Using the equation of motion (s = ut + 0.5at^2), we can solve for time.
distance = 0.5 x acceleration x time^2
3.93 m = 0.5 x acceleration x time^2
Solve for time:
time = √((2 x distance) / acceleration)
Now, let's substitute the given values into the equations to find the solution.