A body of mass 440 grams was
resting on a rough horizontal plane whose coefficient of friction was
one-quarter. The body was dragged across the
plane by the action of a horizontal force. Given that the action of this force
resulted in a uniform acceleration of 170 centimeters per second squared, find the
magnitude of the force. Take the acceleration due to
gravity 𝑔 equal to 9.8 meters per second squared.
We will begin by sketching a
diagram to model the scenario in this question. The body has a mass of 440
grams. This will therefore exert a
downward force of mass multiplied by gravity. If the mass is in kilograms and the
acceleration due to gravity in meters per second squared, the force will be in
newtons. Alternatively, if our mass is
measured in grams and acceleration in centimeters per second squared, the force will
be in dynes. This means that we need to either
convert the 440 grams to kilograms or the 9.8 meters per second squared to
centimeters per second squared.
In this question, we will let 𝑔 be
980 centimeters per second squared as there are 100 centimeters in a meter. The downward force is therefore
equal to 440 multiplied by 980 dynes. Typing this into our calculator
gives us 431,200. By recalling Newton’s third law, we
know there will be a normal reaction force 𝑅 acting in the opposite direction to
this. We are told that there is a
horizontal force 𝐅 acting on the body and it is the magnitude of this force that we
are trying to calculate.
Since the plane is rough, there
will be a frictional force 𝐅 r acting against the motion. We know that the coefficient of
friction 𝜇 is equal to one-quarter. As the body is moving, we know that
the frictional force will be at its maximum. And when this is the case, it is
equal to 𝜇 multiplied by the normal reaction force 𝑅. The frictional force is therefore
equal to one-quarter of the normal reaction force.
The final piece of information we
are given in the question is that the body is moving with a uniform acceleration of
170 centimeters per second squared. In order to calculate the magnitude
of the force 𝐅, we will use Newton’s second law. This states that 𝐅 equals
𝑚𝑎. The sum of the forces is equal to
the mass multiplied by the acceleration.
Resolving vertically, we have the
normal reaction force and the weight force. If we let the positive direction be
vertically upwards, the sum of our forces is 𝑅 minus 431,200. As the body is not accelerating in
this direction, this is equal to zero. Adding 431,200 to both sides of
this equation, we have 𝑅 is equal to 431,200 dynes. Resolving horizontally where the
positive direction is the direction of travel, we have 𝐅 minus 𝐅 r. This is equal to the mass of 440
grams multiplied by the acceleration of 170 centimeters per second squared.
Recalling that the frictional force
𝐅 r is equal to a quarter of the normal reaction force 𝑅, then the left-hand side
simplifies to 𝐅 minus 107,800. Multiplying 440 by 170 gives us
74,800. We can then add 107,800 to both
sides, giving us 𝐅 is equal to 182,600. The magnitude of the force 𝐅 is
therefore equal to 182,600 dynes.