An object of mass 2kg on a plane inclined at angle 40degree and whose coefficient of dynamic friction is 0.3 is connected to another mass of 1kg which hangs vertically over a frictionless pulley. Obtain the acceleration of the system
First find Fn on the 2 kg mass. Net force along an axis perpendicular to the ramp surface is 0, so:
Fn + Fgcos40 = 0
Fn + (-9.81)(2)cos40 = 0
Solve for Fn
Now work out net force of system:
Fa + Fgsin40 + Fk
= 9.81(1) + (-9.81)(2)sin40 + (0.3)(Fn)
Plug in Fn that you found above to get net force of the system.
Divide net force by total mass of system (3kg) to get system acceleration.
To obtain the acceleration of the system, we need to apply Newton's second law of motion. First, let's break down the forces acting on the objects in the system:
1. The force due to gravity on the hanging mass (1 kg):
- This force pulls the hanging mass downward and is given by F_gravity = m * g, where m is the mass of the hanging mass (1 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- Therefore, F_gravity = 1 kg * 9.8 m/s^2 = 9.8 N
2. The force due to the tension in the string:
- Since the hanging mass is connected to the mass on the inclined plane by a string passing over a frictionless pulley, the tension in the string is the same for both masses.
- Since the hanging mass is being pulled downward, the tension in the string pulls the mass on the inclined plane upward.
- Therefore, the force due to tension in the string acting on the mass on the inclined plane is also 9.8 N.
3. The force due to gravity acting on the mass on the inclined plane (2 kg):
- The component of the gravitational force acting parallel to the inclined plane is given by F_parallel = m * g * sin(theta), where theta is the angle of the inclined plane (40 degrees).
- Therefore, F_parallel = 2 kg * 9.8 m/s^2 * sin(40 degrees)
4. The force due to friction acting on the mass on the inclined plane:
- The dynamic friction force can be calculated using the equation F_friction = coefficient of dynamic friction * F_normal, where F_normal is the normal force acting on the object perpendicular to the inclined plane.
- The normal force can be calculated as F_normal = m * g * cos(theta), where theta is the angle of the inclined plane (40 degrees).
- Therefore, F_friction = 0.3 * (2 kg * 9.8 m/s^2 * cos(40 degrees))
Now that we have all the forces acting on the mass on the inclined plane, we can apply Newton's second law:
Sum of forces = mass * acceleration
Considering the positive direction as upward, the sum of forces is given by:
(Force due to tension) - (Force due to friction) - (Force due to gravity parallel to the plane) = (mass on the inclined plane) * (acceleration)
Substituting the known values into the equation:
9.8 N - (0.3 * 2 kg * 9.8 m/s^2 * cos(40 degrees)) - (2 kg * 9.8 m/s^2 * sin(40 degrees)) = (2 kg + 1 kg) * acceleration
Simplifying the equation and solving for acceleration will give us the answer.