A block of Mass 2kg rest on a surface inclined at 30° to the horizontal. Calculate
The normal reaction
The acceleration assuming the surface is smooth
To solve this problem, we will use the concepts of forces and Newton's second law of motion.
1. Calculating the normal reaction:
Since the block is resting on a surface inclined at 30°, the normal reaction force (N) will be perpendicular to the surface. This force counteracts the force of gravity acting on the block, ensuring it remains in equilibrium.
The force of gravity (Fg) acting on the block is given by:
Fg = m * g
where m is the mass of the block (2kg) and g is the acceleration due to gravity (9.8 m/s²).
Fg = 2 kg * 9.8 m/s² = 19.6 N
Since the surface is inclined at 30°, the normal reaction can be calculated using trigonometry. The normal reaction (N) can be resolved into two components: one parallel to the incline (N₁) and one perpendicular to the incline (N₂).
N₂ = N₁ * sin(θ)
N₁ = N / cos(θ)
where θ = 30°.
N = N₁ + N₂
N = N / cos(θ) + N₁ * sin(θ)
The block is in equilibrium, so the sum of the forces in the vertical direction must be zero:
N₂ - Fg = 0
N₁ * sin(θ) - Fg = 0
N₁ * sin(30°) = Fg
N₁ = Fg / sin(30°)
Now we can calculate the normal reaction:
N = N / cos(θ) + N₁ * sin(θ)
N = Fg / sin(30°) / cos(θ) + Fg / sin(30°) * sin(θ)
N = (19.6 N) / (sin(30°) / cos(30°)) + (19.6 N) / sin(30°) * sin(30°)
N = (19.6 N) / (0.5 / 0.866) + 19.6 N
N = 40 N + 19.6 N
N = 59.6 N
Therefore, the normal reaction on the block is 59.6 N.
2. Calculating the acceleration assuming the surface is smooth:
Since the surface is smooth, there is no friction acting on the block. Therefore, the only force acting on the block is the component of gravity in the direction of the incline.
The force parallel to the incline (Fp) can be calculated using trigonometry. Fp is equal to the force of gravity acting on the block multiplied by the sine of the angle of inclination.
Fp = Fg * sin(θ)
Fp = (2 kg * 9.8 m/s²) * sin(30°)
Fp = 2 kg * 9.8 m/s² * 0.5
Fp = 9.8 N
Now, using Newton's second law of motion, we can calculate the acceleration (a) using the equation:
Fp = m * a
a = Fp / m
a = 9.8 N / 2 kg
a = 4.9 m/s²
Therefore, the acceleration of the block assuming the surface is smooth is 4.9 m/s².
To calculate the normal reaction and acceleration, we will use the following equations:
1. The normal reaction (N) can be calculated using the formula: N = mg cos(θ)
2. The acceleration (a) can be calculated using the formula: a = g sin(θ)
Given:
Mass of the block (m) = 2 kg
Inclination angle (θ) = 30°
Step 1: Calculate the gravitational force (weight) acting on the block.
The weight (W) can be calculated using the formula: W = mg
Given:
Mass (m) = 2 kg
Acceleration due to gravity (g) = 9.8 m/s^2
W = 2 kg × 9.8 m/s^2
W = 19.6 N
Step 2: Calculate the normal reaction.
Using the formula: N = mg cos(θ)
N = (2 kg) × (9.8 m/s^2) × cos(30°)
N = 19.6 N × cos(30°)
N = 19.6 N × 0.866
N ≈ 16.98 N
The normal reaction is approximately 16.98 N.
Step 3: Calculate the acceleration.
Using the formula: a = g sin(θ)
a = (9.8 m/s^2) × sin(30°)
a ≈ 4.9 m/s^2 × 0.5
a ≈ 2.45 m/s^2
The acceleration, assuming the surface is smooth, is approximately 2.45 m/s^2.