Mass 1 and mass 2 are connected as shown by a massless, non-stretch rope that passes over a massless, frictionless pulley. The masses are on inclines which have friction. When the angles of the string are as given, mass 1 experiences an acceleration up the incline of 1.00 m/s/s. At that instant, what is the acceleration of mass 2?
m1 = 5.0 kg, m2 = 9.0 kg, θ1 = 25.0
O, θ2 = 35.0 O, Φ1 = 30.0 O, Φ2 = 40.0 O, μ1 = 0.40, μ2 = 0.30,
To find the acceleration of mass 2, we need to analyze the forces acting on both masses.
For mass 1:
1. Calculate the gravitational force acting on mass 1:
F_gravity1 = m1 * g, where g = 9.8 m/s² (acceleration due to gravity).
F_gravity1 = 5.0 kg * 9.8 m/s² = 49.0 N.
2. Calculate the frictional force acting on mass 1:
F_friction1 = μ1 * F_normal1, where μ1 is the coefficient of friction and F_normal1 is the normal force.
F_normal1 = F_gravity1 * cos(θ1), where θ1 is the angle of incline for mass 1.
F_normal1 = 49.0 N * cos(25.0°) = 44.227 N.
F_friction1 = 0.40 * 44.227 N = 17.691 N.
3. Calculate the net force on mass 1:
F_net1 = F_gravity1 * sin(θ1) - F_friction1, where θ1 is the angle of incline for mass 1.
F_net1 = 49.0 N * sin(25.0°) - 17.691 N = 11.159 N.
4. Use Newton's second law to find the acceleration of mass 1:
F_net1 = m1 * a1, where a1 is the acceleration of mass 1.
11.159 N = 5.0 kg * a1.
a1 = 11.159 N / 5.0 kg = 2.232 m/s².
For mass 2:
1. Calculate the gravitational force acting on mass 2:
F_gravity2 = m2 * g, where g = 9.8 m/s² (acceleration due to gravity).
F_gravity2 = 9.0 kg * 9.8 m/s² = 88.2 N.
2. Calculate the frictional force acting on mass 2:
F_friction2 = μ2 * F_normal2, where μ2 is the coefficient of friction and F_normal2 is the normal force.
F_normal2 = F_gravity2 * cos(θ2), where θ2 is the angle of incline for mass 2.
F_normal2 = 88.2 N * cos(35.0°) = 72.637 N.
F_friction2 = 0.30 * 72.637 N = 21.791 N.
3. Calculate the net force on mass 2:
F_net2 = F_gravity2 * sin(θ2) - F_friction2, where θ2 is the angle of incline for mass 2.
F_net2 = 88.2 N * sin(35.0°) - 21.791 N = 30.667 N.
4. Use Newton's second law to find the acceleration of mass 2:
F_net2 = m2 * a2, where a2 is the acceleration of mass 2.
30.667 N = 9.0 kg * a2.
a2 = 30.667 N / 9.0 kg = 3.408 m/s².
Therefore, the acceleration of mass 2 is 3.408 m/s².