A block slides down a frictionless 3m long inclined plane making 10 degrees with the horizontal. The block starts from rest. Find acceleration of the block and its speed at the bottom.
a = g * sin 10 deg
vf^2 = vi^2 + 2 * a *d
To find the acceleration and speed of the block, we can use the principles of physics and trigonometry.
First, let's calculate the acceleration of the block.
The force acting on the block is the component of the gravitational force parallel to the incline. We can calculate this force using the formula:
Force = mass * acceleration
The component of the gravitational force parallel to the incline is given by:
force_parallel = mass * gravity * sin(theta)
Where:
- mass: mass of the block
- gravity: acceleration due to gravity (approximately 9.8 m/s^2)
- theta: angle of inclination (10 degrees, but we need to convert it to radians)
Since the block is on a frictionless surface, no other forces are acting on the block in the direction of motion. Hence, this force is responsible for accelerating the block. Therefore, we have:
force_parallel = mass * acceleration
Substituting the above equation, we get:
mass * acceleration = mass * gravity * sin(theta)
Canceling out the mass from both sides, we find:
acceleration = gravity * sin(theta)
Now, let's calculate the speed of the block at the bottom of the incline. We can use kinematic equations for uniformly accelerated motion.
The equation relating initial velocity (u), final velocity (v), acceleration (a), and distance (s) is:
v^2 = u^2 + 2a * s
Since the block starts from rest (u = 0), we can simplify the equation to:
v^2 = 2a * s
Where:
- u: initial velocity (0 m/s)
- v: final velocity (what we need to find)
- a: acceleration (which we calculated previously)
- s: distance traveled (3 meters)
Substituting the known values, we have:
v^2 = 2 * acceleration * distance
v^2 = 2 * (gravity * sin(theta)) * distance
Finally, taking the square root of both sides, we can find the velocity (speed) at the bottom:
v = sqrt(2 * (gravity * sin(theta)) * distance)
Now, we can plug in the known values and calculate the acceleration and speed:
theta = 10 degrees
Convert theta to radians: theta_rad = theta * (pi/180) = 10 * (pi/180) = 0.1745 radians
distance = 3 meters
acceleration = gravity * sin(theta) = 9.8 * sin(0.1745) ≈ 1.69 m/s^2
v = sqrt(2 * (gravity * sin(theta)) * distance)
v = sqrt(2 * (9.8 * sin(0.1745)) * 3) ≈ 5.64 m/s
Therefore, the acceleration of the block is approximately 1.69 m/s^2, and its speed at the bottom of the incline is approximately 5.64 m/s.