If a frictionless incline has a difference in height from one end to the other of 19.5 cm and has a length along the incline of 1.36 m, find the time it takes for a block of 1.1 kg, starting from rest, to slide down the ramp from the top to the bottom of this incline.
sine of slope angle = .195/1.36 = .143
angle = 8.24 degrees
cos slope angle = .99
weight = 1.1 * 9.81 = 10.8 N
component of weight down slope=10.8(.143)
= 1.54 N
a = F/m = 1.54 / 1.1 = 1.40 m/s^2
d = (1/2) a t^2
1.36 = (1/2)(1.40) t^2
t = 1.39 seconds
To find the time it takes for the block to slide down the incline, we can use the principles of kinematics and energy conservation.
First, we'll calculate the gravitational potential energy (GPE) at the starting position (top of the incline) and at the ending position (bottom of the incline).
The GPE at the top of the incline is given by the equation:
GPE = m * g * h
Where:
m = mass of the block = 1.1 kg
g = acceleration due to gravity = 9.8 m/s^2 (approximately)
h = height difference = 19.5 cm = 0.195 m
So, the GPE at the top is:
GPE(top) = 1.1 kg * 9.8 m/s^2 * 0.195 m
Next, we'll calculate the GPE at the bottom of the incline, which is zero since the block has reached the bottom.
Then, we can equate the GPE at the top with the kinetic energy (KE) at the bottom (assuming no energy is lost):
KE = (1/2) * m * v^2
Where:
m = mass of the block = 1.1 kg
v = velocity of the block at the bottom of the incline
Setting GPE(top) = KE(bottom), we can solve for v:
1.1 kg * 9.8 m/s^2 * 0.195 m = (1/2) * 1.1 kg * v^2
Simplifying:
v^2 = (2 * 9.8 m/s^2 * 0.195 m)
Now, we can find the velocity (v) of the block at the bottom of the incline using the equation:
v = sqrt(2 * g * h)
Substituting the values:
v = sqrt(2 * 9.8 m/s^2 * 0.195 m)
Now that we have the velocity, we can find the time (t) it takes for the block to slide down the incline using the equation of motion:
d = v * t
Where:
d = distance along the incline = 1.36 m
v = velocity of the block at the bottom of the incline (calculated above)
t = time taken
Rearranging the equation:
t = d / v
Substituting the values:
t = 1.36 m / (sqrt(2 * 9.8 m/s^2 * 0.195 m))
Calculating the expression in the denominator, then dividing:
t = 1.36 m / sqrt(3.819 m^2/s^2)
Finally, we can compute the time:
t ≈ 1.36 m / 1.954 m/s ≈ 0.696 s
Therefore, it takes approximately 0.696 seconds for the block to slide down the incline from top to bottom.