A 20 kg mass is allowed to accelerate down a frictionless 15° ramp.
a. Determine the value of the x-component of the force gravity.
b. What is the acceleration of the block down the ramp.
c. How long will it take for the block to slide 30.0m?
To solve this problem, we need to break down the force of gravity into its x and y components. Let's first determine the value of the x-component of the force of gravity.
a. The x-component of the force of gravity is calculated by multiplying the force of gravity (weight) by the cosine of the angle of the ramp.
The formula for the x-component of the force of gravity is given by:
Fx = m * g * cos θ
Where:
m = mass of the object (20 kg)
g = acceleration due to gravity (approx. 9.8 m/s²)
θ = angle of the ramp (15°)
Substituting the values into the formula, we get:
Fx = 20 kg * 9.8 m/s² * cos 15°
Using a calculator, calculate the cosine of 15°:
cos 15° ≈ 0.96592582628
Therefore:
Fx ≈ 20 kg * 9.8 m/s² * 0.96592582628
Calculating this expression:
Fx ≈ 189.134 N
The x-component of the force of gravity is approximately 189.134 N.
b. Since the ramp is frictionless, the acceleration down the ramp will be solely due to the force of gravity. The acceleration is given by:
a = g * sin θ
Using the given values, we calculate the acceleration:
a = 9.8 m/s² * sin 15°
Using a calculator, calculate the sine of 15°:
sin 15° ≈ 0.2588190451
Therefore:
a ≈ 9.8 m/s² * 0.2588190451
Calculating this expression:
a ≈ 2.54 m/s²
The acceleration down the ramp is approximately 2.54 m/s².
c. To determine the time it takes for the block to slide 30.0 m, we can use the equation of motion:
s = ut + (1/2)at²
where:
s = distance (30.0 m)
u = initial velocity (0 m/s, as the block starts from rest)
t = time
a = acceleration (2.54 m/s²)
Rearranging the equation, we get:
t = sqrt((2s) / a)
Substituting the values:
t = sqrt((2 * 30.0 m) / 2.54 m/s²)
Calculating this expression:
t ≈ sqrt(59.06) ≈ 7.68 s
Therefore, it will take approximately 7.68 seconds for the block to slide 30.0 m down the ramp.
To determine the value of the x-component of the force of gravity, we need to consider the angle of the ramp. The force of gravity can be broken down into two components: the x-component (along the ramp) and the y-component (perpendicular to the ramp).
a. To find the x-component of the force gravity, we need to calculate the gravitational force acting on the mass. The formula for the force of gravity is given by:
Fgravity = m · g
Where:
m = mass of the object (20 kg)
g = acceleration due to gravity (9.8 m/s^2)
Fgravity = 20 kg · 9.8 m/s^2 = 196 N
The x-component of the force of gravity can be calculated using the formula:
Fx = Fgravity · sin(θ)
Where:
θ = angle of the ramp (15°)
Fx = 196 N · sin(15°) = 49.91 N
Therefore, the x-component of the force of gravity is approximately 49.91 N.
b. To calculate the acceleration of the block down the ramp, we can use the formula:
acceleration = Fx / mass
acceleration = 49.91 N / 20 kg = 2.495 m/s^2
Therefore, the acceleration of the block down the ramp is approximately 2.495 m/s^2.
c. To determine the time it will take for the block to slide 30.0 m, we can use the kinematic equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
At the top of the ramp, the initial velocity is 0 m/s since the block starts from rest.
distance = (1/2) * acceleration * time^2
Rearranging the equation:
time = √(2 * distance / acceleration)
time = √(2 * 30 m / 2.495 m/s^2) = √(120.48) ≈ 10.97 s
Therefore, it will take approximately 10.97 seconds for the block to slide 30.0 m down the ramp.
a) zero. Gravity acts downward. That is usually called the y direction. The Force component DOWN the ramp is
M g sin 15.
What are they calling the x direction?
b) a = g sin15 = 2.54 m/s^2
c) t = sqrt(2L/a) = 4.86 s