a stick of linear density p=2kg/m rests on a disc of radius=1m. the stick makes an angle of 60° with horizontal and is tangent to the disc at its upper end .Friction exists at all points of contact ,and assume that it is large enough to keep the system at rest . Find friction force between ground and disc.
To find the friction force between the ground and the disc, we can follow these steps:
Step 1: Calculate the weight of the stick
The weight of the stick can be calculated using the formula: weight = mass * gravitational acceleration.
The mass of the stick can be determined by integrating the linear density over its length.
Given that the linear density, p, is 2 kg/m, and the length of the stick is l, we can use the formula:
Mass of the stick = ∫p dx from 0 to l
= ∫2 dx from 0 to l
= 2 * x from 0 to l
= 2 * l kg
Step 2: Calculate the normal force on the disc
The normal force exerted on the disc is equal to the weight of the stick acting in the vertical direction, since the system is at rest.
Normal force = weight of the stick = 2 * l * gravitational acceleration
Step 3: Calculate the frictional force
The frictional force between the ground and the disc is equal in magnitude but opposite in direction to the horizontal component of the normal force on the disc.
Frictional force = normal force * coefficient of friction
Since the system is at rest, the frictional force balances the horizontal component of the normal force:
Frictional force = 2 * l * gravitational acceleration * coefficient of friction
Therefore, the friction force between the ground and the disc is equal to 2 * l * gravitational acceleration * coefficient of friction.
To find the friction force between the ground and the disc, we need to analyze the forces acting on the system and apply Newton's laws of motion. Here are the steps to find the friction force:
Step 1: Identify the forces:
In this system, there are three main forces:
1. Weight of the stick: This is the force due to gravity acting on the stick. It can be calculated using the formula W = mg, where m is the mass of the stick and g is the acceleration due to gravity.
2. Normal force from the disc: This force acts perpendicular to the surface of the disc at the point of contact with the stick.
3. Friction force between the ground and the disc: This is the force we need to find. It acts parallel to the surface of the ground and opposes the tendency of motion.
Step 2: Resolve the weight force:
Since the stick makes an angle of 60° with the horizontal, we need to resolve the weight force into components. The vertical component contributes to the normal force, while the horizontal component contributes to the friction force.
The vertical component of the weight force can be calculated using the formula mg cos θ, where θ is the angle between the stick and the horizontal.
The horizontal component of the weight force can be calculated using the formula mg sin θ.
Step 3: Solve for the normal force:
At the point of contact between the stick and the disc, the normal force is equal in magnitude and opposite in direction to the vertical component of the weight force. Therefore, the normal force is mg cos θ.
Step 4: Solve for the friction force:
The maximum value of the static friction force is μN, where μ is the coefficient of static friction between the ground and the disc, and N is the magnitude of the normal force. Since the system is at rest, the static friction force must equal the horizontal component of the weight force.
So, the friction force between the ground and the disc is equal to mg sin θ.
Step 5: Substitute the given values and solve:
Given:
- Linear density of the stick: λ = 2 kg/m
- Radius of the disc: r = 1 m
- Angle between the stick and horizontal: θ = 60°
To find the mass of the stick, we need to calculate the length of the stick:
- The stick is tangent to the disc at its upper end, forming a right triangle.
- The hypotenuse of the triangle is the sum of the radius of the disc and the length of the stick.
- Using trigonometry, we can find the length of the stick as (1 + 2√3) m.
The mass of the stick can be calculated by multiplying the linear density by the length of the stick: m = λ * L.
Now, substitute the given values into the formulas for the weight force and the friction force to find the answers.
Note: Make sure to convert the angle from degrees to radians when performing trigonometric calculations.
By following these steps, you can find the friction force between the ground and the disc.