Mass M is initially sliding down the inclined plane with a velocity of V = 4 cm/sec as shown. The coefficient of kinetic friction is μK = 0.2. Angle of 70 degrees. Assuming the mass does not reach the base of the incline before one second has passed, answer the following:
(a) Draw Free Body Diagram of the mass.
(b) What is the speed of the mass at t = 1 sec?
(c) How far from its original position has the mass slid down the plane after t = 1 sec?
(a) To draw the free body diagram of the mass, we need to consider all the forces acting on it.
1. Weight (mg): This force acts vertically downwards and its magnitude is given by the formula Fg = mg, where m is the mass of the object and g is the acceleration due to gravity.
2. Normal force (N): This force acts perpendicular to the incline and prevents the mass from sinking into the plane. Its magnitude is equal to the component of the weight perpendicular to the incline, which is given by N = mg * cos(theta), where theta is the angle of the incline.
3. Friction force (f): This force opposes the motion of the mass and acts parallel to the incline. Its magnitude is given by f = μK * N, where μK is the coefficient of kinetic friction.
4. Component of the weight parallel to the incline (mg * sin(theta)) which acts downwards.
The free body diagram should show the mass (M) with these forces labeled and indicated in the appropriate directions. Make sure the angle of the incline (70 degrees) is correctly shown as well.
(b) To find the speed of the mass at t = 1 sec, we need to use kinematic equations. Since the acceleration due to gravity acts vertically downwards and is not parallel to the incline, we need to resolve it into components parallel and perpendicular to the incline.
The component of gravity parallel to the incline is given by g * sin(theta), and the component perpendicular to the incline is g * cos(theta). The parallel component acts downwards and provides an acceleration in the direction of motion.
Using the equation of motion for constant acceleration, we have:
Vf = Vi + a * t
where Vf is the final velocity, Vi is the initial velocity (given as 4 cm/sec), a is the acceleration, and t is the time (1 sec).
Since there is no external force acting parallel to the incline, the acceleration is equal to the parallel component of gravity, which is g * sin(theta).
Vf = 4 + (g * sin(theta)) * t
Substitute the known values of g = 9.8 m/s^2 and theta = 70 degrees (convert to radians) into the equation and calculate Vf.
(c) To find how far the mass has slid down the plane after t = 1 sec, we can use the equation of motion for constant acceleration:
S = Vi * t + (1/2) * a * t^2
where S is the displacement, Vi is the initial velocity (given as 4 cm/sec), a is the acceleration (parallel component of gravity), and t is the time (1 sec).
Plugging in the known values, calculate the displacement S.