Finding Force and Angle for Mop Head Movement

[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.

 

[tiny-tim]

Hi intenzxboi!

(have a theta: θ and a phi: φ )

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

 

[nlightner]

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.