Blocks 1 & 2 are connected by a string. Block 1 is on an incline & has a mass of 5.4kg. Block 2 is hanging from the string & has a mass of 2.1kg. The coefficient of friction is 0.22 between all surfaces.
1: What is the minimum angle "theta" can be while keeping the blocks motionless?
2: What is the maximum angle "theta" can be while keeping the blocks motionless?
Given:
M1 = 5.4kg
M2 = 2.1kg
Max. Theta = ?
Min. Theta = ?
Friction = 0.22
To find the minimum and maximum angles "theta" that keep the blocks motionless, we need to consider the forces acting on the system.
Let's break down the forces acting on each block:
For Block 1 (on the incline):
- Gravity force (mg) pulls the block downwards.
- The normal force (N) is perpendicular to the surface of the incline.
- The friction force (f) opposes the motion and acts up the incline.
- The component of gravity force (mg sin(theta)) pulls the block downhill.
- The component of gravity force (mg cos(theta)) presses the block into the incline.
For Block 2 (hanging from the string):
- Gravity force (mg) pulls the block downwards.
- The tension in the string (T) acts upwards.
Now, let's analyze the forces to find the minimum and maximum angles:
1. Minimum angle "theta":
In this case, the minimum angle occurs when the force of static friction (f) equals its maximum value. This means that the block is on the verge of moving but is still motionless.
Using the formula for static friction: f = μN, where μ is the coefficient of friction and N is the normal force, we can determine the normal force acting on Block 1:
N = mg cos(theta)
Then, substitute N into the expression for static friction:
f = μ(mg cos(theta))
The static friction will be at its maximum when it equals the force component downhill (mg sin(theta)) acting on Block 1:
f = mg sin(theta)
By setting these two expressions equal to each other, we can solve for the minimum angle:
μ(mg cos(theta)) = mg sin(theta)
Simplifying and solving for theta, we have:
tan(theta) = μ
Substituting the given coefficient of friction (μ = 0.22), we find:
theta = tan^(-1)(0.22)
2. Maximum angle "theta":
The maximum angle occurs when the force of static friction is at its minimum, allowing the block to remain motionless.
Since the static friction is counteracting the force component downhill (mg sin(theta)), the force of static friction can be given as:
f = -mg sin(theta)
Here, we need to calculate the maximum force of static friction (f_max) using the formula: f_max = μN. Using the expression for the normal force (N = mg cos(theta)), we find:
f_max = μ(mg cos(theta))
The maximum angle will be the angle where the force of static friction (f) reaches its maximum value (f_max). Therefore, we have:
mg sin(theta) = μ(mg cos(theta))
Dividing both sides by mg and simplifying, we get:
tan(theta) = 1/μ
Substituting the given coefficient of friction (μ = 0.22), we find:
theta = tan^(-1)(1/0.22)
So, the minimum angle "theta" is approximately tan^(-1)(0.22), and the maximum angle "theta" is approximately tan^(-1)(1/0.22).