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12 - Bonus assignment for Newton's 2nd Law and friction
Due: 8:00pm on Wednesday, February 29, 2012
Note: To understand how points are awarded, read your instructor's Grading Policy.
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This is an extra credit problem. Your grade will not go down if you don't work on it. However, you can earn bonus credit if you do.
Pushing a Lawnmower
Consider a lawnmower of mass
handle, which makes an angle
which can slide across a horizontal surface with a coefficient of friction
with the horizontal. Assume that
. In this problem the lawnmower is pushed using a massless
, 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
for the magnitude of the acceleration due to gravity.
Part A
Find the magnitude,
Hint A.1
, of the force required to slide the lawnmower over the ground at constant speed by pushing the handle.
How to approach this problem
.
If the lawnmower is to move at constant velocity, the net force acting on it must be zero. Using this fact, find two equations, one for the vertical and one
for the horizontal components of all forces acting on the lawnmower. Solve these equations for
, eliminating unknown quantities in favor of quantities
given in the problem introduction.
Hint A.2
Compute the sum of vertical forces
The lawnmower is not accelerating vertically. Express the sum of the vertical components of all the forces on the lawnmower.
ANSWER:
=
Answer Requested
Hint A.3
Find
=0
Compute the normal force
, the magnitude of the normal force exerted on the lawnmower by the ground.
Express
in terms of
ANSWER:
=
Do not assume that
and other given variables.
Correct
without starting from
. Here's an example of one of the many ways that assumption can fail--when
there is another force on the object that has a vertical component.
Hint A.4
Compute the sum of horizontal forces
The phrase "at constant speed" implies that there is no x component of acceleration. Therefore, the sum of the x components of all the real forces
acting on the lawnmower is zero. Find the sum of the x components all of of the forces. Remember, the positive direction is up and to the right.
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4/16/2012
MasteringPhysics: Assignment Print View
Express your answer in terms of the given forces, , and
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, the magnitude of the friction force.
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ANSWER:
Answer Requested
Once you have obtained the equations for the vertical and horizontal equilibrium, solve them simultaneously to find
.
Express the required force in terms of given quantities.
ANSWER:
=
Answer Requested
Part B
The solution for
has a singularity (that is, becomes infinitely large) at a certain angle
However, a negative applied force
. For any angle
, the expression for
will be negative.
would reverse the direction of friction acting on the lawnmower, and thus this is not a physically acceptable solution. In fact, the
increased normal force at these large angles makes the force of friction too large to move the lawnmower at all.
Find an expression for
Hint B.1
.
How to approach the problem
Look at the expression you have derived for
in Part A. The force
will become infinite when the denominator becomes zero. Use this criterion to
find the required quantity.
ANSWER:
=
Answer Requested
You should have found that
, the force required to push the lawnmower at constant speed, was
.
Note that this expression becomes infinite when the denominator equals zero:
,
or
.
(The phrase "
has a singularity at angle
" means that "
goes to infinity at a certain angle
It's not too hard to understand what this means. Suppose you were pushing straight down on the lawnmower (
according to the equation for
, when you plug in
.")
degrees). It obviously wouldn't move. But,
degrees, you get a negative force (which doesn't make sense).
The more vertical you push, the harder it gets to move the lawnmower. At
, it gets impossible to move it. The force required to move it goes to infinity;
you have to push infinitely hard.
Score Summary:
Your score on this assignment is 10.5%.
You received 2.1 out of a possible total of 20 points.
http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1738195
4/16/2012