A 23-kg mass starts at rest at the bottom of a 40 degrees frictionless incline. A force of 300 N is applied to the mass directed up the incline. a) What is the acceleration of the mass up the incline ? b) If the top of the incline has a height of 5 m above the ground, how long will it take the mass to reach the top of the incline?
To answer these questions, we need to apply the principles of Newton's second law and the equations of motion.
a) To find the acceleration of the mass up the incline, we can use the following equation:
F_net = m * a
where F_net is the net force acting on the mass, m is the mass, and a is the acceleration. In this case, the net force is the force applied up the incline (300 N), the mass is given as 23 kg, and we need to find the acceleration.
First, we need to resolve the force applied up the incline into its components. The force can be divided into two perpendicular components: the force parallel to the incline and the force perpendicular to the incline. Since the incline is frictionless, there is no force perpendicular to the incline.
The force parallel to the incline can be found using trigonometry. The component of the applied force that acts parallel to the incline is given by:
F_parallel = F_applied * sin(theta)
where theta is the angle of the incline (40 degrees).
F_parallel = 300 N * sin(40 degrees)
= 300 N * 0.6428
≈ 192.85 N
Now, we can substitute this value into the equation for net force:
F_net = F_parallel = m * a
192.85 N = 23 kg * a
Solving for a:
a = 192.85 N / 23 kg
≈ 8.38 m/s²
So, the acceleration of the mass up the incline is approximately 8.38 m/s².
b) To calculate the time it takes for the mass to reach the top of the incline, we can use the equations of motion. Since the mass starts from rest, we can use the following equation:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s since it comes to rest at the top), u is the initial velocity (0 m/s), a is the acceleration (8.38 m/s²), and s is the distance traveled along the incline (the height of the incline, 5 m).
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
Plugging in the given values:
s = (0^2 - 0^2) / (2 * 8.38 m/s²)
= 0 / 16.76 m/s²
= 0 m
Since the distance traveled along the incline is 0 m, meaning the mass doesn't entirely reach the top of the incline, we cannot calculate the time it takes to reach the top.
Therefore, the mass will not reach the top of the incline.