A block is released from rest at the top of an inclined plane. The height of the plane is 5.00 m and it makes an angle of 30.0 degrees with the horizontal . The coefficient of kinetic friction between block and surface is 0.200. What is the speed of the block (in m/s) when it reaches the bottom of the incline?
Horizontal direction :
IF angle is measured from y axis (vertical line) -> a(horizontal) = Fsin(32)/m
IF angle is measured from horizontal -> a(horizontal) = Fcos(32)/m
Vertical direction:
[ here positive a means acceleration down, negative a means up]
IF angle is measured from vertical:
If force is down -> a(vertical) = (Fcos(32) + mg)/m
if force is up -> a(vertical) = (Fcos(32) - mg)/m
If angle is measured from horizontal
force is down -> a = (Fsin(32) + mg)/m
force is up -> F = (Fsin(32) - mg)/m
Then magnitude of acceleration = SQRT[acceleration(horizontal) ^ 2 + acceleration(vertical) ^ 2]
To find the speed of the block when it reaches the bottom of the incline, we can use the principles of mechanical energy conservation.
First, let's calculate the potential energy (PE) of the block at the top of the incline using the formula: PE = mgh, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the incline.
PE = mgh
PE = (mass)(gravity)(height)
PE = (m)(9.8 m/s²)(5.00 m)
Next, we'll calculate the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance traveled. The force of friction can be found using the formula: force of friction = coefficient of friction × normal force. The normal force can be determined by taking the component of the weight perpendicular to the incline, which is given by: normal force = weight × cos(θ), where θ is the angle of the incline.
normal force = (mass)(gravity) × cos(θ)
Once we have the force of friction, we can calculate the work done by friction as: work done by friction = force of friction × distance = force of friction × (length of the incline).
Next, we'll calculate the change in potential energy. At the bottom of the incline, the potential energy is zero.
Finally, we'll use the principle of mechanical energy conservation to calculate the kinetic energy (KE) at the bottom of the incline. The total mechanical energy (ME) remains constant, so ME = KE + PE. Rearranging the equation, we get: KE = ME - PE.
Since the block starts from rest, the initial kinetic energy is zero. Therefore, the final kinetic energy at the bottom of the incline is equal to the total mechanical energy. We can then use the formula for kinetic energy: KE = 0.5mv², where m is the mass of the block and v is the speed of the block.
Finally, we can solve for v by substituting the calculated values for the mass, height, angle, and friction coefficient into the equations and solving for v.