A 100 kg mass starts from rest at the top of an incline plane that makes an angle of 35 degrees
with the horizontal. There is a coefficient of friction of 0.005 between the ramp and the mass.
The mass slides down the ramp and then slides to a stop on a surface that the coefficient of
friction is 0.9, between the mass and the surface. ( 72 Points, 6 Each )
a.) How much work is done by the gravitational force while the mass is on the incline?
b.) How much work is done by the normal force while the mass is on the incline?
c.) How much work is done by the frictional force while the mass is on the incline?
d.) What is the net work ( sum of the work ) done while the mass is on the incline?
e.) What is the net force down the ramp?
f.) How much work was done by the net force while the mass was on the incline?
g.) How far does the mass slide along the horizontal surface before coming to rest?
h.) How much work is done by the gravitational force during that slide?
I.) How much work is done by the normal force during that slide?
j.) How much work is done by the frictional force during that slide?
k.) What is the net work done on the mass during that slide?
l.) What is the change in kinetic energy of the block during that slide?
To solve these questions, we need to apply the principles of work and energy.
a) To find the work done by the gravitational force, we need to determine the vertical component of the force. The vertical component is given by the formula:
F_gravity = m * g * sin(theta),
where m represents the mass, g is the acceleration due to gravity (approx. 9.8 m/s^2), and theta is the angle of the incline (35 degrees).
The work done by the gravitational force can be calculated using the formula:
Work_gravity = F_gravity * d,
where d represents the distance moved along the incline.
b) The normal force is perpendicular to the incline and does not do any work because it is perpendicular to the displacement. Therefore, the work done by the normal force is zero.
c) The work done by the frictional force is given by the formula:
Work_friction = frictional_force * d,
where frictional_force = coefficient_of_friction * normal_force, and d represents the distance moved along the incline.
d) The net work done on the mass is equal to the sum of the work done by the gravitational force and the work done by the frictional force. So, we can calculate it by:
Net Work = Work_gravity + Work_friction.
e) The net force down the ramp is the component of the gravitational force parallel to the incline minus the frictional force:
Net force = (m * g * sin(theta)) - (coefficient_of_friction * m * g * cos(theta)).
f) The work done by the net force is given by:
Work_net = Net force * d.
g) To find the distance the mass slides along the horizontal surface, we need to apply the principle of conservation of energy. The work done by the net force is equal to the change in kinetic energy of the mass.
Since the mass comes to a stop, the change in kinetic energy is zero. Therefore, the work done by the net force is zero. From part f, we found the work done by the net force. By equating this to zero, we can solve for d.
h) Since the mass comes to a stop, the work done by the gravitational force during that slide is equal to the negative of the work done by the net force. So:
Work_gravity_slide = -Work_net.
i) The normal force does not do any work during the horizontal slide because it is perpendicular to the displacement. Therefore, the work done by the normal force is zero.
j) The work done by the frictional force during the slide is given by:
Work_friction_slide = frictional_force * d,
where frictional_force = coefficient_of_friction_slide * normal_force_slide, and d represents the distance moved along the horizontal surface.
k) The net work done on the mass during the slide is equal to the sum of the work done by the gravitational force during the slide and the work done by the frictional force during the slide. So:
Net Work_slide = Work_gravity_slide + Work_friction_slide.
l) The change in kinetic energy of the block during the slide is given by:
Change in Kinetic Energy = Net Work_slide.
To solve these questions, you will need to calculate the value of sin(theta), cos(theta), normal_force, coefficient_of_friction_slide, and normal_force_slide using the given information. Then you can substitute these values into the formulas and solve for the respective quantities in each question.