A physics student pulls a block of mass m = 20 kg up an incline at a slow constant velocity for a distance of d = 4.5 m. The incline makes an angle Δ = 30° with the horizontal. The coefficient of kinetic friction between the block and the inclined plane is µk = 0.2.
a)What is the work Wm done by the student?
b)At the top of the incline, the string by which she was pulling the block breaks. The block, which was at rest, slides down a distance d = 4.5 m before it reaches a friction less horizontal surface. A spring is mounted horizontally on the friction less surface with one end attached to a wall. The block hits the spring, compresses it a distance L = 0.6 m, then rebounds back from the spring, retraces its path along the horizontal surface, and climbs up the incline.
What is the speed v of the block when it first reaches the horizontal surface?
c)What is the spring constant k of the spring?
d)How far up the incline d1 does the block rebound?
a) To find the work done by the student, we need to calculate the work done against the force of friction and the work done against gravity.
First, we calculate the work done against the force of friction. The force of friction can be calculated using the formula:
Frictional Force = (coefficient of kinetic friction) * (normal force)
The normal force can be calculated using the formula:
Normal Force = (mass of the block) * (acceleration due to gravity) * (cosine of the angle of the incline)
Next, we calculate the work done against gravity. The force of gravity can be calculated using the formula:
Force of Gravity = (mass of the block) * (acceleration due to gravity) * (sine of the angle of the incline)
The work done against gravity is given by the formula:
Work Against Gravity = (force of gravity) * (distance)
Finally, the total work done by the student is given by:
Total Work = Work Against Friction + Work Against Gravity
b) To find the speed of the block when it first reaches the horizontal surface, we need to use the conservation of mechanical energy. The block initially has gravitational potential energy and no kinetic energy. When it reaches the bottom of the incline, it will have kinetic energy and no potential energy.
Using the principle of conservation of mechanical energy, we can set the initial gravitational potential energy equal to the final kinetic energy:
Initial Gravitational Potential Energy = Final Kinetic Energy
The initial gravitational potential energy can be calculated using the formula:
Initial Gravitational Potential Energy = (mass of the block) * (acceleration due to gravity) * (height of the incline)
The final kinetic energy can be calculated using the formula:
Final Kinetic Energy = (1/2) * (mass of the block) * (velocity squared)
Setting these two equations equal to each other and solving for velocity will give us the speed of the block when it first reaches the horizontal surface.
c) To find the spring constant k of the spring, we can use Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
The force exerted by the spring can be calculated using the formula:
Force = (spring constant) * (displacement)
Since we know the displacement (0.6 m) and the force exerted by the spring, we can rearrange Hooke's Law to solve for the spring constant:
Spring Constant = Force / Displacement
d) To find how far up the incline the block rebounds, we need to use the conservation of mechanical energy again. The block starts with kinetic energy at the bottom of the incline and compressed potential energy in the spring. When it reaches the top of the incline, it will have gravitational potential energy and no kinetic energy.
Using the principle of conservation of mechanical energy again, we can set the initial kinetic energy equal to the final gravitational potential energy:
Initial Kinetic Energy = Final Gravitational Potential Energy
The initial kinetic energy can be calculated using the formula:
Initial Kinetic Energy = (1/2) * (mass of the block) * (velocity squared)
The final gravitational potential energy can be calculated using the formula:
Final Gravitational Potential Energy = (mass of the block) * (acceleration due to gravity) * (height of the incline - rebound distance)
Setting these two equations equal to each other and solving for the rebound distance will give us how far up the incline the block rebounds.
M*g = 20 * 9.8 = 196 N. = Wt. of block.
Fp = 196*sin30 = 98 N. = Force parallel
t0 the incline.
Fn = 196*Cos30 = 169.7 N. = Normal force
Fk = u*Fn = 0.2 * 169.7 = 33.95 N. = Force of kinetic friction.
a. Fap-Fp-Fk = M*a
a = 0
Fap = Force applied.
Solve for Fap.
Work = Fap * d
d = 4.5 m.