A box of mass 10.9 kg with an initial velocity
of 2.01 m/s slides down a plane, inclined
at 38◦ with respect to the horizontal. The coefficient
of kinetic friction is 1. The box stops
after sliding a distance x. The acceleration
due to gravity is 9.8 m/s.
a. How far does the box slide?. The positive
x-direction is down the plane.
Answer in units of m
b.What is the the work done by friction?
Answer in units of J
c.What is the work done by the normal force?
Answer in units of J
d.What is the magnitude of the work done by gravity?
Answer in units of J
e.What is the magnitude of the instantaneous
power generated by friction half way between
the initial and final positions?
Answer in units of W
f.What is the magnitude of the average power
generated by friction from start to stop?
Answer in units of W
a. Well, to find out how far the box slides, we need to use some good old-fashioned physics. The formula we can use is the work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy. In this case, the box comes to a stop, so its change in kinetic energy is zero. The work done by friction is given by the equation:
Work = Force of friction x distance
Now, the force of friction can be calculated using the formula:
Force of friction = coefficient of kinetic friction x normal force
The normal force can be calculated using the equation:
Normal force = mass x g x cos(angle)
where g is the acceleration due to gravity.
We know the mass of the box, the coefficient of kinetic friction, the acceleration due to gravity, and the angle of the plane. So, let's plug in the values and calculate the distance.
b. To find the work done by friction, we can use the formula:
Work = Force of friction x distance
But we already know the force of friction (calculated in part a) and the distance. So, plug in the values and calculate the work done.
c. The work done by the normal force is zero. Why? Because the normal force acts perpendicular to the direction of motion, and the definition of work is force multiplied by the displacement in the direction of the force. Since the force and displacement are perpendicular, the work done by the normal force is zero.
d. The magnitude of the work done by gravity can be calculated using the formula:
Work = m x g x height
where m is the mass of the box, g is the acceleration due to gravity, and height is the vertical distance the box slides. Since the box slides down the inclined plane, we can calculate the vertical distance using the formula:
height = distance x sin(angle).
Plug in the values and calculate the work done by gravity.
e. To calculate the magnitude of the instantaneous power generated by friction halfway between the initial and final positions, we need to know the instantaneous speed at that point. Unfortunately, we don't have that information, so we can't calculate it.
f. The magnitude of the average power generated by friction from start to stop can be calculated using the formula:
Power = Work / time
Since we don't have the time it takes for the box to stop, we can't calculate the average power generated by friction.
I hope this helps! Let me know if you have any more questions, and I'll do my best to bring the humor.
To solve this problem, we will use various kinematic and work-energy equations. Let's go through each part step-by-step.
a. How far does the box slide?
To find the distance traveled by the box, we need to use the equations of motion. The acceleration of the box is given by the component of gravity along the incline:
a = g*sinθ = 9.8 m/s^2 * sin(38°)
Using the equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (zero since the box stops)
u = initial velocity (2.01 m/s)
a = acceleration along the incline
s = distance traveled
Rearranging the equation, we get:
s = (v^2 - u^2) / (2a)
Substituting the known values:
s = (0^2 - 2.01^2) / (2 * a)
Calculate the value of a and substitute it:
s = (0 - 2.01^2) / (2 * 9.8 * sin(38°))
Now you can calculate the value of s to get the distance traveled by the box.
b. What is the work done by friction?
The work done by friction can be calculated using the equation:
Work = force * displacement
The frictional force can be found using:
frictional force = coefficient of friction * normal force
Given that the coefficient of kinetic friction is 1 and the normal force is equal to the weight of the box, which is mg:
frictional force = 1 * m * g
The work done by friction is equal to the force of friction multiplied by the distance traveled (s):
Work = frictional force * s
Substitute the known values and calculate the work done by friction.
c. What is the work done by the normal force?
The work done by the normal force is zero because the normal force acts perpendicular to the displacement, and the work done by a force is zero if it is perpendicular to the displacement.
d. What is the magnitude of the work done by gravity?
The work done by gravity is equal to the product of the displacement and the component of the gravitational force in the direction of displacement.
Work by gravity = m * g * s * cos(θ)
Substitute the known values and calculate the work done by gravity.
e. What is the magnitude of the instantaneous power generated by friction halfway between the initial and final positions?
To find the instantaneous power generated by friction, we need to find the force of friction (same as calculated in part b) and the instantaneous velocity at the midpoint.
The force of friction is given by:
frictional force = coefficient of kinetic friction * normal force
The instantaneous velocity at the midpoint can be found by dividing the initial velocity by 2.
Once you have the force of friction and the instantaneous velocity, you can calculate the power using the equation:
Power = force * velocity
Substitute the known values and calculate the instantaneous power.
f. What is the magnitude of the average power generated by friction from start to stop?
The average power generated by friction can be calculated using the equation:
Average Power = Work / Time
Since the box stops, the time taken is the time it takes for the box to slide down the incline.
Using the equation:
Time = sqrt(2 * s / a)
Substitute the known values and calculate the average power generated by friction.
To solve these problems, we need to use the concepts of work, friction, and power.
a. To find the distance the box slides, we can use the equation of motion that relates distance, initial velocity, acceleration, and time:
s = ut + 1/2 * a * t^2
In this case, the initial velocity is 2.01 m/s, the acceleration is the component of gravity down the plane, and the time taken to stop is not given. However, we can find the time using the fact that the box stops when it reaches zero velocity. The deceleration will be equal to the kinetic friction acting on the box.
The frictional force can be calculated using:
f_friction = friction coefficient * normal force
Where the normal force is equal to the weight of the box. Hence:
f_friction = 1 * (mass * gravity)
Once we have the frictional force, we can find the deceleration:
deceleration = frictional force / mass
Now, to calculate the time taken to stop, we can use the equation of motion:
final velocity = initial velocity + acceleration * time
Since final velocity is zero, we can solve for time:
0 = 2.01 + (-deceleration) * time
Solving for time, we get:
time = 2.01 / deceleration
Finally, we can substitute this value of time into the equation of motion for distance to find the distance the box slides:
s = 2.01 * (2.01 / deceleration) + 1/2 * deceleration * (2.01 / deceleration)^2
s = 2.01^2 / (2 * deceleration) + (2.01^2 / (2 * deceleration))
s = 2 * (2.01^2) / (2 * deceleration)
s = (2.01^2) / deceleration
Now, we can substitute the value of deceleration calculated earlier to find the distance s.
b. The work done by friction can be calculated using the equation:
work = force * distance
In this case, the force of friction is equal to the frictional force (calculated earlier) and the distance is the same as the distance the box slides. Therefore, the work done by friction is given by:
work_friction = frictional force * distance
c. The work done by the normal force is zero because the normal force acts perpendicular to the displacement, and no component of the normal force is in the direction of motion.
d. The work done by gravity can be calculated using the equation:
work_gravity = force_gravity * distance
The force of gravity acting on the box is the weight of the box, which is equal to the mass of the box multiplied by the acceleration due to gravity.
e. The instantaneous power generated by friction halfway between the initial and final positions can be calculated using the formula:
power = force * velocity
In this case, the force of friction is the same as the frictional force (calculated earlier), and the velocity is the average velocity between the initial and final velocities. Since the box starts from rest and stops, the average velocity halfway is given by half of the initial velocity.
f. The average power generated by friction from start to stop can be calculated using the formula:
average power = work / time
In this case, the work done by friction is the same as the work done by friction (calculated earlier), and the time is the same as the time taken to stop the box.
M*g = 10.9kg * 9.8N./kg = 106.82 N. = Wt
of the box.
Fp = 106.82*sin38 = 65.8 N. = Force parallel to the incline.
Fn = 106.82*Cos38 = 84.2 N. = Normal
force.
Fk = u*Fn = 1 * 84.2 = 84.2 N. = Force of kinetic friction.
Note: If the coefficient of kinetic friction is 1, the box cannot slide down the incline; because the kinetic
friction is higher than the force
parallel to the incline(Fp).