A rectangular block of mass10kg rests on an inclined plane with a coefficient of static friction of0.55 and a coefficient of kinetic friction is 0.25 at what angle will the box begin to slide and it’s acceleration
To find the angle at which the box will begin to slide, we need to compare the force of static friction with the force of gravity component parallel to the inclined plane.
The force of gravity component parallel to the inclined plane is given by:
F_parallel = mg sin(theta)
where m is the mass of the block (10 kg), g is the acceleration due to gravity (9.8 m/s^2), and theta is the angle of the incline.
The force of static friction can be found using the formula:
F_friction = u_static * N
where u_static is the coefficient of static friction (0.55) and N is the normal force. The normal force can be found using the formula:
N = mg cos(theta)
Putting it all together, the force of static friction is:
F_friction = u_static * N = u_static * mg cos(theta)
Now, for the box to begin sliding, the force of static friction must be equal to the force parallel to the inclined plane:
F_friction = F_parallel
u_static * mg cos(theta) = mg sin(theta)
Dividing both sides by mg:
u_static * cos(theta) = sin(theta)
Rearranging:
u_static / sin(theta) = cos(theta)
Taking the inverse tangent of both sides:
tan^(-1)(u_static / sin(theta)) = theta
Using the given coefficient of static friction (0.55):
theta = tan^(-1)(0.55 / sin(theta))
Using a numerical method or trial and error, we find that the value of theta is approximately 30.96 degrees.
To find the acceleration of the block once it starts sliding, we can use the formula:
F_friction = u_kinetic * N
where u_kinetic is the coefficient of kinetic friction (0.25).
The force of friction during sliding is:
F_friction = u_kinetic * N = u_kinetic * mg cos(theta)
The force parallel to the inclined plane is still:
F_parallel = mg sin(theta)
The net force acting on the block is given by:
F_net = F_parallel - F_friction
Using Newton's second law (F = ma), the acceleration of the block is:
a = F_net / m = (mg sin(theta) - u_kinetic * mg cos(theta)) / m
Substituting the given values and the previously calculated theta:
a = (10 kg)(9.8 m/s^2)(sin(30.96°) - 0.25 cos(30.96°)) / 10 kg
Calculating this expression gives us an acceleration of approximately 2.89 m/s^2.