Two blocks of weights 77.0 and 114.0 N are connected by as massless string and slide down an inclined plane making an angle of 45.0 deg. to the horizontal. The cofficient of kinetic friction between the lighter block and the plane is 0.15 and that between the heavier block and the plane is 0.23. Assuming that the lighter block ``leads'', find the magnitude of the heavier block's acceleration.
The blocks move with the same acceleration ‘a’
F(fr)= μN
m₁g=W₁ => m₁=W₁/g= 77/9.8=7.86 kg
m₂g=W₂ => m₂=W₂/g= 114/9.8=11.63 kg
x: m₁a=m₁g•sin α -μ₁N₁-T
y: 0=-m₁g•cos α +N₁
x: m₂a=m₂g•sin α -μ₂N₂+T
y: 0=-m₂g•cos α +N₂
m₁a=m₁g•sin α -μ₁•m₁g•cos α - T
m₂a=m₂g•sin α -μ₂•m₂g•cos α +T
a= g•[sin α(m₁+m₂) - cosα(μ₁•m₁+ μ₂•m₂)]/(m₁+m₂)
how do u find u1?
To find the magnitude of the heavier block's acceleration, we can break down the problem into two parts: the forces acting on the lighter block and the forces acting on the heavier block.
Let's start with the forces on the lighter block:
1. Gravitational force (Fg1): This force acts vertically downward and is equal to the weight of the lighter block. The weight can be calculated by multiplying the mass (77.0 N) by the acceleration due to gravity (9.8 m/s^2):
Fg1 = 77.0 N * 9.8 m/s^2 = 754.6 N
2. Normal force (Fn1): This force acts perpendicular to the incline. Since the block is sliding down the incline, the normal force is less than the weight. The normal force can be calculated using the formula:
Fn1 = Fg1 * cos(45.0°)
3. Frictional force (Ff1): This force acts parallel to the incline and opposes the motion of the lighter block. The frictional force can be calculated using the formula:
Ff1 = coefficient of kinetic friction1 * Fn1
Now let's move on to the forces on the heavier block:
1. Gravitational force (Fg2): This force acts vertically downward and is equal to the weight of the heavier block. The weight can be calculated by multiplying the mass (114.0 N) by the acceleration due to gravity (9.8 m/s^2):
Fg2 = 114.0 N * 9.8 m/s^2 = 1117.2 N
2. Normal force (Fn2): This force acts perpendicular to the incline. Similarly to the lighter block, the normal force is less than the weight since the block is sliding down the incline. The normal force can be calculated using the formula:
Fn2 = Fg2 * cos(45.0°)
3. Frictional force (Ff2): This force acts parallel to the incline and opposes the motion of the heavier block. The frictional force can be calculated using the formula:
Ff2 = coefficient of kinetic friction2 * Fn2
Now that we have calculated all the forces, we can find the net force acting on the system. The net force is the difference between the gravitational force (down the incline) and the frictional force (opposite to the motion).
Net force = (Fg1 - Ff1) - (Fg2 - Ff2)
Using Newton's second law (F = m * a), we can relate the net force to the acceleration of the system. Since the blocks are connected by a massless string, they will have the same acceleration.
Net force = (77.0 N * a) + (114.0 N * a)
Simplifying the equation:
Net force = 191.0 N * a = (Fg1 - Ff1) - (Fg2 - Ff2)
Finally, we can solve for the acceleration (a):
a = (Fg1 - Ff1 - (Fg2 - Ff2)) / 191.0 N
Plug in the values of Fg1, Ff1, Fg2, Ff2, and the coefficients of kinetic friction1 and 2 to calculate the magnitude of the heavier block's acceleration.