Abox of mass 60kg starts from the rest at height h and slides down a rough slope of length 10m, which makes an angle of 25 degree with the horizontal. It undergoes a constant accelequation of magnitude 2m/ while sliding down the slope. Calculate the work done on the box by the frictional force, using the work-energy theorem
Wb = m*g = 60kg * 9.8N/kg = 588 N. = Wt.
of box.
Fp = 588*sin25 = 248.5 N. = Force
parallel to the slope.
Fp-Fk = m*a
248.5-Fk = 60 * 2 = 120
Fk = 248.5 - 120 = 128.5 N. = Force of
kinetic friction.
Work = Fk * L = 128.5 * 10 = 1285 Joules
Suppose you use the principle of work done by friction is equal to the change in the kinetic energy of the body. what would be the mistake?
To calculate the work done on the box by the frictional force, we need to calculate the work done by the gravitational force and the work done by the applied force.
1. Work done by the gravitational force:
The gravitational force is given by the formula:
F_gravity = m * g
where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The displacement along the slope is given by:
s = h * sin(theta)
where h is the initial height and theta is the angle with respect to the horizontal.
The work done by the gravitational force is given by:
W_gravity = F_gravity * s * cos(theta)
= m * g * h * sin(theta) * cos(theta)
2. Work done by the applied force:
The applied force is responsible for the acceleration of the box.
The work done by the applied force can be calculated using the work-energy theorem.
The work done by the applied force is given by:
W_applied = (1/2) * m * v^2
where v is the final velocity of the box when it reaches the bottom of the slope.
Using the kinematic equation:
v^2 = u^2 + 2 * a * s
where u is the initial velocity (which is zero since the box starts from rest), a is the constant acceleration along the slope, and s is the displacement along the slope.
Plugging in the values, we get:
v^2 = 0 + 2 * 2 * h * sin(theta)
= 4 * h * sin(theta)
Substituting this value in the equation for work done by the applied force, we get:
W_applied = (1/2) * m * (4 * h * sin(theta))
= 2 * m * h * sin(theta)
3. Work done on the box by the frictional force:
According to the work-energy theorem, the net work done on the box is equal to its change in kinetic energy.
Since the box starts from rest, its initial kinetic energy is zero.
Therefore, the net work done on the box is equal to its final kinetic energy.
The final kinetic energy of the box is given by:
K_final = (1/2) * m * v^2
Substituting the value of v^2, we get:
K_final = (1/2) * m * (4 * h * sin(theta))
= 2 * m * h * sin(theta)
This is the same expression as the work done by the applied force.
So, the work done on the box by the frictional force is zero.
Therefore, the work done on the box by the frictional force is zero.
To calculate the work done on the box by the frictional force, we need to first calculate the work done by the gravitational force as the box slides down the slope, and then use the work-energy theorem.
1. Calculate the work done by the gravitational force:
The work done by a force can be calculated using the formula:
Work = Force x Distance x cos(theta)
where:
- Force is the component of the force in the direction of motion
- Distance is the distance over which the force acts
- theta is the angle between the force vector and the displacement vector
In this case, the force is the gravitational force acting on the box, given by the formula:
Force = Mass x Acceleration due to gravity
The distance over which the force acts is the length of the slope, which is 10m.
The angle theta between the force vector (vertical) and the displacement vector (along the slope) can be calculated as:
theta = atan(slope)
Given that the slope makes an angle of 25 degrees with the horizontal, the slope can be calculated as:
slope = tan(theta) = tan(25 degrees)
Substituting these values into the formula, we can calculate the work done by the gravitational force.
2. Use the work-energy theorem to calculate the work done by the frictional force:
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Since the box starts from rest, its initial kinetic energy is zero. Therefore, the work done by the frictional force is equal to the total work done on the box.
Now, let's go step-by-step to calculate the work done by the gravitational force and then use the work-energy theorem to find the work done by the frictional force.
Step 1: Calculate the slope:
slope = tan(25 degrees)
Step 2: Calculate the gravitational force:
Force = mass x acceleration due to gravity
Force = 60kg x 9.8m/s^2
Step 3: Calculate the angle theta:
theta = atan(slope)
Step 4: Calculate the work done by the gravitational force:
Work_gravity = Force x distance x cos(theta)
Work_gravity = (60kg x 9.8m/s^2) x 10m x cos(theta)
Step 5: Calculate the work done by the frictional force using the work-energy theorem:
Work_friction = Work_total - Work_gravity
Work_friction = Work_total - (60kg x 9.8m/s^2 x 10m x cos(theta))
Where Work_total is the total work done on the box.
By following these steps and substituting the appropriate values, you can calculate the work done on the box by the frictional force using the work-energy theorem.