# Finding Force and Angle for Mop Head Movement

• intenzxboi
In summary, the conversation discusses a worker pushing a mop with a force at an angle of 40° with the vertical, and the coefficients of static and kinetic friction between the mop head and the floor. The question asks for the force needed for the mop head to move with a constant velocity and the angle at which the force is unable to move the mop head. A hint is given to put the equation in a different form to solve for the angle.
intenzxboi

## Homework Statement

In Figure 6-62 a fastidious worker pushes directly along the handle of a mop with a force . The handle is at an angle θ = 40° with the vertical, and μs = 0.61 and μk = 0.48 are the coefficients of static and kinetic friction between the head of the mop and the floor. Ignore the mass of the handle and assume that all the mop's mass m = 0.65 kg is in its head. (a) If the mop head moves along the floor with a constant velocity, then what is F? (b) If θ is less than a certain value θ0, then (still directed along the handle) is unable to move the mop head. Find θ0.

Im having trouble understanding how to solve for part b.

i got to:
Fsin0 - Fs (mg-Fcos0)=0

but i can't solve of the angle.

Hi intenzxboi!

(have a theta: θ and a phi: φ )

Hint: put Asinθ + Bcosθ in the form C(sinθcosφ + cosθsinφ), for some angle φ

I would approach this problem by first reviewing the given information and identifying the key variables and equations involved. From the problem statement, we can see that the force applied is along the handle of the mop, and the angle between the handle and the vertical is 40°. We are also given the coefficients of friction for the mop head and the floor, as well as the mass of the mop head.

For part a, we are asked to find the force (F) required to maintain a constant velocity of the mop head. We can use the equation for the net force on an object to solve for F:

Fnet = F - μmgcosθ = 0

where F is the applied force, μ is the coefficient of kinetic friction, m is the mass of the mop head, g is the acceleration due to gravity, and θ is the angle between the handle and the vertical. Solving for F, we get:

F = μmgcosθ

Plugging in the given values, we get:

F = (0.48)(0.65 kg)(9.8 m/s^2)cos40°

F = 2.99 N

Therefore, the required force to maintain a constant velocity for the mop head is 2.99 N.

For part b, we are asked to find the critical angle (θ0) at which the applied force is unable to move the mop head. In other words, the force applied is equal to the maximum static friction force, and any increase in the applied force will cause the mop head to start moving. We can use the equation for the maximum static friction force to solve for θ0:

Fmax = μsmgcosθ0

where μs is the coefficient of static friction. Plugging in the given values, we get:

F = (0.61)(0.65 kg)(9.8 m/s^2)cosθ0

F = 3.75cosθ0

To solve for θ0, we need to find the value of cosθ0 that makes the maximum static friction force equal to the applied force. This can be done by setting the two equations for F equal to each other and solving for cosθ0:

μsmgcosθ0 = μmgcos40°

(0.61)(0.65 kg)(9.8 m/s^2)cosθ0 = (0.48)(0.

## 1. How do you calculate the force and angle for mop head movement?

The force and angle for mop head movement can be calculated using the formula F = ma, where F is the force, m is the mass of the mop head, and a is the acceleration. The angle can be determined using trigonometric functions such as sine, cosine, and tangent.

## 2. What factors affect the force and angle for mop head movement?

The force and angle for mop head movement can be affected by various factors such as the weight and type of the mop head, the surface it is being used on, and the force applied by the person using the mop.

## 3. How can the force and angle for mop head movement be measured?

The force can be measured using a force meter or scale, while the angle can be measured using a protractor or by using trigonometric functions with the known sides of the triangle formed by the mop head movement.

## 4. Why is it important to find the force and angle for mop head movement?

Calculating the force and angle for mop head movement is important because it helps in understanding the mechanics of how a mop works and how to use it efficiently. It can also help in determining the amount of force needed for different surfaces and the best angle for effective cleaning.

## 5. Are there any safety precautions to consider when finding the force and angle for mop head movement?

Yes, it is important to always use caution when handling and using a mop to avoid injury. It is also important to use the correct amount of force and angle to prevent strain or damage to the mop or the surface being cleaned.

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