A block of mass m = 5.2 kg is pulled up a è = 24° incline as in the figure with a force of magnitude F = 34 N.
a) Find the acceleration of the block if the incline is frictionless.
b)Find the acceleration of the block if the coefficient of kinetic friction between the block and incline is 0.12.
a)
F=ma
F-mg sin 24= ma
a= 34-[ 5.2(9.81) sin 24]/5.2
a=2.55m/s^2
b)
F=ma
mg sin 24-Fk=ma
(5.2x 9.81x sin 24)- (5.2x 9.81x cos 24)(0.12)= 5.2a
20.75-5.59=5.2a
a=2.91m/s^2
To find the acceleration of the block up the incline, we need to first resolve the force of 34 N into its components parallel and perpendicular to the incline.
The force component parallel to the incline (F_parallel) will help in determining the force responsible for the block's acceleration, while the component perpendicular to the incline (F_perpendicular) will not affect the block's motion along the incline.
The force component parallel to the incline can be found using the formula:
F_parallel = F * sin(è)
Substituting the given values:
F_parallel = 34 N * sin(24°)
Now, we can calculate the force responsible for the block's acceleration (F_responsible) by subtracting the frictional force (f) acting in the opposite direction:
F_responsible = F_parallel - f
To find the frictional force, we can use the formula:
f = μ * N
where μ is the coefficient of friction and N is the normal force. The normal force is equal to the weight of the block, which can be calculated as:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values:
N = 5.2 kg * 9.8 m/s^2
Once we have the normal force, we can calculate the frictional force using the coefficient of friction provided in the problem.
Finally, substitute the values of F_parallel and f into the equation for F_responsible:
F_responsible = F_parallel - f
With F_responsible calculated, we can find the acceleration (a) using Newton's second law:
F_responsible = m * a
Divide both sides of the equation by m:
a = F_responsible / m
Substitute the values of F_responsible and m to find the acceleration of the block up the incline.