Final answer:
To find the acceleration, we use the formula a = g * sin(θ); to determine the friction force, we use the formula f = u * N; to calculate the tension in the rope, we use the equation T = mg + ma; and to identify the velocity, we use the equation v^2 = u^2 + 2as.
Step-by-step explanation:
To find the answers to the given questions, we need to utilize the concepts of inclined planes and Newton's second law of motion.
A) Find acceleration:
Using the formula a = g * sin(θ), where g is the acceleration due to gravity (9.8 m/s^2) and θ is the incline angle (30 degrees), we can calculate the acceleration. Plugging in the values, we get:
a = 9.8 m/s^2 * sin(30 degrees)
a ≈ 4.9 m/s^2
B) Determine friction force:
To determine the friction force, we can use the formula f = u * N, where u is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the cart, which is mg.
f = u * mg
Plugging in the values, we have:
f = 0.20 * 20 kg * 9.8 m/s^2
f ≈ 39.2 N
C) Calculate tension in the rope:
To calculate the tension in the rope, we need to consider the forces acting on the cart. The tension acts in the opposite direction of the cart's motion, counteracting the gravitational force. Therefore, the tension T is equal to the sum of the gravitational force (mg) and the force due to acceleration (ma).
T = mg + ma
Plugging in the values, we get:
T = 20 kg * 9.8 m/s^2 + 20 kg * 4.9 m/s^2
T = 392 N
D) Identify the velocity:
To find the velocity, we can use the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which is assumed to be 0 in this case), a is the acceleration, and s is the distance/height traveled.
Plugging in the values, we have:
v^2 = 0 + 2 * 4.9 m/s^2 * s
v^2 = 9.8 m/s^2 * s
Since the sled comes to rest after traveling 80 m along the inclined plane, we can solve for v:
0 = 9.8 m/s^2 * 80 m
v ≈ 38.98 m/s