Find the magnitude, , of the force required to slide the lawnmower over the ground at constant speed by pushin

Question

Consider a lawnmower of weight which can slide across a horizontal surface with a coefficient of friction . In this problem the lawnmower is pushed using a massless handle, which makes an angle with the horizontal. Assume that , the force exerted by the handle, is parallel to the handle.Take the positive x direction to be to the right and the postive y direction to be upward.Find the magnitude, , of the force required to slide the lawnmower over the ground at constant speed by pushing the handle.

Answer 1

Draw a force diagram.

You have some kinetic friction pushing in the negative x direction on the lawnmower. You also have a force along the lawnmower handle pointed at the lawnmower at an angle with the horizontal. You also have the weight of the lawnmower pushing striaght down in the y direction. Finally, you have some normal force pushing up (in the y direction) on the lawnmower from the ground.

Since you are not accelerating (i.e., you're at constant velocity) in the x direction, the forces in the x direction must sum to zero. Thus, you have in the x direction:

F cos(theta) - Ff = 0

where F is the force down the lawnmower handle and theta is the angle the handle makes with the horizontal; Ff is the force of friction. So F*cos(theta) is the x component of the handle force. Thus, you want to pick F so that:

F cos(theta) = Ff

However, we don't know Ff so we have to look at the y direction to find it. Since you're not moving in the y direction, your acceleration is constant, and so again forces sum to 0 and you have:

F sin(theta) + W - Fn = 0

where, again, F is the handle force and theta the angle it makes with the horizontal. Fn is the normal force from the ground. W is the weight of the lawnmower. Here F*sin(theta) is the y component of the handle force. From this second equation it is clear that:

Fn = F*sin(theta) + W

You also know the friction force is:

Ff = mu*Fn = mu*(F*sin(theta) + W)

where mu is the coefficient of kinetic friction. Thus, you want to solve for F such that F*cos(theta) = Ff, but we know the form of Ff, so we want to solve for F so that:

F cos(theta) = mu*(F*sin(theta) + W) = mu*F*sin(theta) + mu*W

You can pull the terms with F to one side to get:

F cos(theta) - mu*F*sin(theta) = mu*W

and then you can factor the F out:

F( cos(theta) - mu*sin(theta) ) = mu*W

and then divide by the factor with the trig functions

F = mu*W/( cos(theta) - mu*sin(theta) )

And that's your solution. If you push with that force, you will move at a constant speed. Note that if there is no friction, mu will be equal to 0, and you will only move at a constant speed if the force is 0.

Note that the denominator ( cos(theta) - mu*sin(theta) ) must be STRICTLY GREATER THAN 0. It cannot be equal to 0 or less than zero. If it is equal to or less than 0, then the frictional force will be greater than the applied force and the lawnmower will slow to a stop and not move. This should make sense to you. Imagine that your handle was straight up in the air at 90 degrees. In this case, cos(theta)=0 and sin(theta)=1. Clearly, (cos(theta)-mu*sin(theta))<0 regardless of choice of mu. In this condition, all of your force goes straight into the ground and none goes toward pushing the lawnmower. If the lawnmower was already moving, your extra force would simply increase the friction force and the lawnmower would come to a stop. If it was already stopped, it would stay stopped.

So the angle should be picked such that:

(cos(theta)-mu*sin(theta)) > 0

and, in order to travel at a constant forward speed, F should be picked such that:

F = mu*W/( cos(theta) - mu*sin(theta) )

where W is the weight of the lawnmower, which is (mass of lawnmower)*(acceleration due to gravity).

Answer 2

In outer area an merchandise at consistent speed has not something interfering with it. on the earth, an merchandise shifting at a relentless speed will decelerate because of the fact of different forces interfering. This includes, friction, rolling resistance, (the two considerable issues), bumps interior the line, wind resistance and different issues. it could look that consistent rigidity is mandatory to maintain consistent speed because of the fact we don't word the different forces interacting with the speed.

Answer 3

A magnitude of 43.

Answer 4

Once you apply the force you dont have to push it any more.
But if you apply power to the mower you are continuously uning energy which result in its motion.
Note you cannot push a massless handle.

Answer 5

u find it, do ur own homework