Two blocks A and B are connected by a light in extensible cord passing over a frictionless pulley. Block A starts from rest and moves up the smooth plane which is inclined at 30 degrees to the horizontal. Calculate at the moment that A has moved 4 m along the plane.
(a) the total kinetic energy of the system
(b) the speed of the blocks A and B
To answer these questions, we can break them down into smaller steps:
Step 1: Calculate the acceleration of block A.
Since the plane is inclined at 30 degrees, we can use trigonometry to find the acceleration of block A. The component of the gravitational force acting parallel to the inclined plane is given by F_parallel = m*g*sin(theta), where m is the mass of block A and g is the acceleration due to gravity. The net force acting on block A is equal to the force parallel to the inclined plane, so we have F_net = m*a, where a is the acceleration. Therefore, m*a = m*g*sin(theta), and we can solve for a.
Step 2: Calculate the time it takes for block A to move 4 m.
Using the formula s = ut + (1/2)*a*t^2, where s is the distance moved, u is the initial velocity, and t is the time, we can rearrange the equation to solve for t: t^2 + 2u*t - (2s/a) = 0. Here, since block A starts from rest, its initial velocity u is 0, and we can solve for t.
Step 3: Calculate the velocity of block A.
Using the equation v = u + at, we can plug in the values of u (0), a (from step 1), and t (from step 2) to calculate the velocity of block A.
Step 4: Calculate the velocity of block B.
Since block A is connected to block B by the light inextensible cord, their velocities are the same. Therefore, the velocity of block B is equal to the velocity of block A calculated in step 3.
Step 5: Calculate the total kinetic energy of the system.
The total kinetic energy of the system is the sum of the kinetic energies of block A and block B. The kinetic energy of an object is given by the formula KE = (1/2)*m*v^2, where m is the mass of the object and v is its velocity. Substitute the values for mass and velocity of each block into the formula and add them together to find the total kinetic energy of the system.
By following these steps, you can calculate the total kinetic energy of the system and the speed of blocks A and B when block A has moved 4 m along the inclined plane.
The picture is not complete.
Are the two blocks identical (in mass) ?
Does block B hang free?
If it is, B will descend as A moves up.
Draw the free body diagram of each block.
Let the tension in string be T.
mass of each block = m
acceleration due to gravity = g
angle of incline = θ = 30°
Distance moved = s = 4m
Change in potential energy of system, δep
mgs(sinθ)-mgs
=mgs(sinθ-1)
Assuming no frictional and air-resistance loss,
δep+δek=0, or
δek=mgs(1-sinθ)
Equate δek to (1/2)(m+m)v² to solve for v.