A 275 kg piano slides 4.2 m down a 30° incline and is kept from accelerating by a man who is pushing back on it parallel to the incline. The effective coefficient of kinetic friction is 0.40
(a) Calculate the force exerted by the man.
N
(b) Calculate the work done by the man on the piano.
J
(c) Calculate the work done by the friction force.
J
(d) What is the work done by the force of gravity?
J
(e) What is the net work done on the piano?
J
a. M*g = 275*9.8 = 2695 N. = Wt. of piano.
Fp = 2695*sin30 = 1348 N. = Force parallel with plane.
Fn = 2695*Cos30 = 2334 N. = Normal force.
u*Fn = 0.4*2334 = 934 N. = Force of kinetic friction.
Fe-Fp+u*Fn = M*a.
Fe-1348+934 = M*0,
Fe - 414 = 0,
Fe = 414 N. = Force exerted.
b. W = Fe*d = 414 * 4.2 = 1739 Joules.
c. W = uFn * d = 934 * 4.2 =
d. W = 1348 * 4.2 =
component of weight down slope = 275 * 9.81 * sin 30 = 1349 N
component of weight normal to slope = 275*9.81*cos 30 = 2336 N
so friction force up = .4 *2336 = 935 N
total force up = F + 935
total force down = 1349
no acceleration so force up = force down slope
F + 935 = 1349
solve for F in Newtons
now work = force * distance
F is up, motion is down so work done by man is -4.2 F Newton meters or Joules
The friction force, same deal
However gravity is component down slope times motion down slope so positive 1349 * 4.2
well gravity work better be equal and opposite to sum of man and friction :)
e. Fnet = Fe-Fp+uFn = 414-1348+934 = 0.
Wnet = Fnet * d = 0 * 4.2 = 0.
To find the answers, we need to use the concepts of force, work, and displacement. Here's how we can approach each part of the question:
(a) The force exerted by the man is equal in magnitude but opposite in direction to the force of friction. We can start by calculating the force of friction using the equation:
Force of friction = coefficient of kinetic friction * Normal force
The normal force is the perpendicular component of the weight of the piano, which can be calculated as:
Normal force = Weight * cos(angle of incline)
Using the given information, the weight of the piano is:
Weight = mass * gravity
Now we can calculate the normal force and then substitute it into the equation for the force of friction to find the force exerted by the man.
(b) The work done by the man on the piano can be calculated using the equation:
Work = Force * displacement * cos(theta)
Where theta is the angle between the force and displacement vectors. In this case, the angle is 0 degrees since the force is parallel to the displacement.
(c) The work done by the friction force can be calculated using the equation:
Work by friction = Force of friction * displacement * cos(theta)
Again, the angle theta is 0 degrees because the force of friction is parallel to the displacement.
(d) The work done by the force of gravity can be calculated using the equation:
Work by gravity = Weight * displacement * cos(theta)
In this case, the angle theta is 180 degrees because the force of gravity acts in the opposite direction of the displacement.
(e) The net work done on the piano is the sum of the work done by all the forces acting on it. We can calculate it by adding the work done by the man, the work done by the friction force, and the work done by gravity.
Now that we have outlined the steps, let's calculate the values:
(a) Calculating the force exerted by the man:
- Calculate the weight of the piano: Weight = mass * gravity
- Calculate the normal force: Normal force = Weight * cos(angle of incline)
- Calculate the force of friction: Force of friction = coefficient of kinetic friction * Normal force
- The force exerted by the man is equal in magnitude but opposite in direction to the force of friction.
(b) Calculating the work done by the man:
- Use the equation: Work = Force * displacement * cos(theta)
(c) Calculating the work done by the friction force:
- Use the equation: Work by friction = Force of friction * displacement * cos(theta)
(d) Calculating the work done by the force of gravity:
- Use the equation: Work by gravity = Weight * displacement * cos(theta)
(e) Calculating the net work done on the piano:
- Sum up the work done by the man, the work done by the friction force, and the work done by gravity.
By following these steps, you should be able to find the numerical values for each part of the question.