Consider a lawnmower of mass m which can slide across a horizontal surface with a coefficient of friction μ. In this problem the lawnmower is pushed using a massless handle, which makes an angle θ with the horizontal. Assume that Fh, 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. Use g for the magnitude of the acceleration due to gravity.

Part A

Find the magnitude, Fh, of the force required to slide the lawnmower over the ground at constant speed by pushing the handle.

Express the required force in terms of given quantities.

Fh =

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Part B

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Fh = u*m*g / (cos (θ) - u*sin (θ))

Explanation:

Given:

- The mass of lawnmower = m

- The angle the handle makes with the horizontal = θ

- The force applied along the handle = Fh

- The coefficient of friction of the lawnmower with ground = u

Find:

Find the magnitude, Fh, of the force required to slide the lawnmower over the ground at constant speed by pushing the handle.

Solution:

- Construct a Free Body Diagram (FBD) for the lawnmower.

- Realize that there is horizontal force applied parallel to ground due to Fh that drives the lawnmower and a friction force that opposes this motion. We will use to Newton's law of motion to express these two forces in x-direction as follows:

F_net, x = m*a

- Since, the lawnmower is to move with constant speed then we have a = 0.

F_net, x = 0

- The forces as follows:

Fh*cos (θ) - Ff = 0

Where, Ff is the frictional force:

Fh = Ff / cos (θ)

Similarly, for vertical direction y the forces are in equilibrium. Using equilibrium equation in y direction we have:

- W - Fh*sin (θ) + Fn = 0

Where, W is the weight of the lawnmower and Fn is the contact force exerted by the ground on the lawnmower. Then we have:

Fn = W + Fh*sin (θ)

Fn = m*g + Fh*sin (θ)

The Frictional force Ff is proportional to the contact force Fn by:

Ff = u*Fn

Ff = u * (m*g + Fh*sin (θ))

Substitute this expression in the form derived for Fh and Ff:

Fh*cos (θ) = u * (m*g + Fh*sin (θ))

Fh * (cos (θ) - u*sin (θ)) = u*m*g

Fh = u*m*g / (cos (θ) - u*sin (θ))