A smooth body slides down a plane inclined at an angle of 60degree to the horizontal calculate, A. The components of the acceleration of the body. B. Along the inclined plane. C. Perpendicular to the plane. D. The velocity of the body at the base of the plane after 10s. E. The vertical height of fall.
lemon
To calculate the components of the acceleration of the body when it slides down the inclined plane at an angle of 60 degrees to the horizontal, we can use the following equations:
A. Components of the acceleration:
The acceleration of the body can be split into two components: one along the inclined plane and another perpendicular to the plane.
1. Acceleration along the inclined plane:
The acceleration along the inclined plane (a_parallel) can be determined using the equation:
a_parallel = g * sin(θ)
where g is the acceleration due to gravity (approximately 9.8 m/s²) and θ is the angle of the incline.
So, for θ = 60 degrees:
a_parallel = 9.8 * sin(60)
= 9.8 * 0.866
≈ 8.5 m/s²
2. Acceleration perpendicular to the plane:
The acceleration perpendicular to the plane (a_perpendicular) is equal to zero since gravity does not act in that direction.
B. Along the inclined plane:
The acceleration along the inclined plane is 8.5 m/s² (as calculated above).
C. Perpendicular to the plane:
The acceleration perpendicular to the plane is 0 m/s².
D. Velocity of the body at the base of the plane after 10 seconds:
To calculate the velocity of the body at the base of the plane after 10 seconds, we can use the following equation:
v = u + a * t
Given:
Initial velocity (u) = 0 m/s (assuming the body starts from rest)
Acceleration (a) = 8.5 m/s²
Time (t) = 10 s
Using the equation:
v = 0 + 8.5 * 10
= 85 m/s
So, the velocity of the body at the base of the plane after 10 seconds is 85 m/s.
E. Vertical height of fall:
To calculate the vertical height of fall, we can use the following formula:
h = (u² * sin²(θ)) / (2 * g)
Given:
Initial velocity (u) = 0 m/s (as the body starts from rest)
Angle of inclination (θ) = 60 degrees
Acceleration due to gravity (g) = 9.8 m/s²
Using the equation:
h = (0² * sin²(60)) / (2 * 9.8)
= 0 / 19.6
= 0 m
Therefore, the vertical height of fall is zero.
To calculate the components of the acceleration of the body on an inclined plane, we can use basic trigonometry and the equations of motion. Let's go step by step:
A. Components of acceleration:
The acceleration can be divided into two components: one parallel to the inclined plane and the other perpendicular to it.
1. Acceleration along the inclined plane (a_parallel):
We can calculate this component using the equation a_parallel = g * sin(θ), where g is the acceleration due to gravity (approximately 9.8 m/s²) and θ is the angle of inclination (60 degrees).
So, a_parallel = 9.8 m/s² * sin(60°) = 8.48 m/s² (rounded to two decimal places).
2. Acceleration perpendicular to the inclined plane (a_perpendicular):
This component is zero since there is no force directly perpendicular to the inclined plane.
B. Acceleration along the inclined plane:
We have already calculated this component in part A. It is a_parallel = 8.48 m/s².
C. Acceleration perpendicular to the inclined plane:
This component is zero, as mentioned earlier.
D. Velocity of the body at the base of the plane after 10 seconds:
To find the velocity at the base of the plane after 10 seconds, we need to consider the initial velocity, which is assumed to be zero.
We can use the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
In this case, u = 0 m/s, a = a_parallel = 8.48 m/s² (from part A), and t = 10 seconds.
v = 0 + 8.48 m/s² * 10 s = 84.8 m/s (rounded to one decimal place).
E. Vertical height of fall:
To find the vertical height of fall, we can use the concept of potential energy. The initial potential energy (at the top of the inclined plane) is equal to the final kinetic energy (at the base of the plane).
The potential energy at the top of the inclined plane is given by m * g * h, where m is the mass of the body and h is the height.
The kinetic energy at the base of the plane is given by (1/2) * m * v², where v is the final velocity calculated in part D.
Since the body slides down without friction, we can assume that there is no change in mechanical energy, so the potential energy at the top equals the kinetic energy at the base.
m * g * h = (1/2) * m * v²
The mass cancels out from both sides of the equation:
g * h = (1/2) * v²
So, h = (1/2) * v² / g
Substituting the values, h = (1/2) * (84.8 m/s)² / 9.8 m/s² = 366.4 m (rounded to one decimal place).
Therefore, the vertical height of fall is 366.4 meters.