Determine the normal force for a laundry basket with a mass of 5.8 kg in each of the following situations
(a) at rest on a horizontal surface
(b) at rest on a ramp inclined at 12° above the horizontal
(c) at rest on a ramp inclined at 25° above the horizontal
(d) at rest on a ramp inclined at 45° above the horizontal
To determine the normal force for each situation, we need to consider the forces acting on the laundry basket.
(a) When at rest on a horizontal surface, the normal force is equal to the gravitational force acting on the basket. Therefore, the normal force is given by:
Normal force = Weight = mass × gravitational acceleration
Given:
Mass of the laundry basket (m) = 5.8 kg
Gravitational acceleration (g) = 9.8 m/s^2
Normal force = 5.8 kg × 9.8 m/s^2 = 56.84 N
Therefore, the normal force on the laundry basket when at rest on a horizontal surface is 56.84 N.
(b) When at rest on a ramp inclined at 12° above the horizontal, we need to consider the gravitational force acting vertically downwards and the component of the weight acting perpendicular to the ramp.
The normal force can be found using the formula:
Normal force = Weight × cos(angle)
Given:
Mass of the laundry basket (m) = 5.8 kg
Gravitational acceleration (g) = 9.8 m/s^2
Angle of the ramp (θ) = 12°
Weight = mass × gravitational acceleration
Weight = 5.8 kg × 9.8 m/s^2
Normal force = Weight × cos(θ)
Normal force = (5.8 kg × 9.8 m/s^2) × cos(12°)
Normal force ≈ 56.51 N
Therefore, the normal force on the laundry basket when at rest on a ramp inclined at 12° above the horizontal is approximately 56.51 N.
(c) When at rest on a ramp inclined at 25° above the horizontal, we use the same formula as in part (b) to calculate the normal force:
Normal force = Weight × cos(θ)
Given:
Mass of the laundry basket (m) = 5.8 kg
Gravitational acceleration (g) = 9.8 m/s^2
Angle of the ramp (θ) = 25°
Weight = mass × gravitational acceleration
Weight = 5.8 kg × 9.8 m/s^2
Normal force = Weight × cos(θ)
Normal force = (5.8 kg × 9.8 m/s^2) × cos(25°)
Normal force ≈ 51.77 N
Therefore, the normal force on the laundry basket when at rest on a ramp inclined at 25° above the horizontal is approximately 51.77 N.
(d) When at rest on a ramp inclined at 45° above the horizontal, we again use the same formula as in part (b) to calculate the normal force:
Normal force = Weight × cos(θ)
Given:
Mass of the laundry basket (m) = 5.8 kg
Gravitational acceleration (g) = 9.8 m/s^2
Angle of the ramp (θ) = 45°
Weight = mass × gravitational acceleration
Weight = 5.8 kg × 9.8 m/s^2
Normal force = Weight × cos(θ)
Normal force = (5.8 kg × 9.8 m/s^2) × cos(45°)
Normal force ≈ 40.96 N
Therefore, the normal force on the laundry basket when at rest on a ramp inclined at 45° above the horizontal is approximately 40.96 N.
To determine the normal force acting on the laundry basket in each of the given situations, we need to consider the forces acting on the basket.
(a) When the basket is at rest on a horizontal surface, the normal force (N) will be equal to the weight of the basket (mg), where m is the mass and g is the acceleration due to gravity. In this case, N = mg.
(b) When the basket is at rest on a ramp inclined at 12° above the horizontal, we need to consider the weight of the basket and its component perpendicular to the ramp. The normal force (N) will counterbalance this perpendicular component of weight to keep the basket in equilibrium.
We can break the weight (mg) into two components: the perpendicular component (mg * cosθ) and the parallel component (mg * sinθ), where θ is the angle of the ramp with respect to the horizontal. The normal force, which is equal to the perpendicular component of weight, can be calculated as N = mg * cosθ.
(c) Similar to situation (b), we need to consider the perpendicular component of weight to find the normal force when the basket is at rest on a ramp inclined at 25° above the horizontal. Here, N = mg * cosθ.
(d) In this situation, the ramp is inclined at 45° above the horizontal, which means it is at its steepest point. The normal force (N) will solely balance the weight of the basket. So, N = mg.
Now, let's calculate the normal forces for each situation:
(a) N = mg = 5.8 kg * 9.8 m/s² = 56.84 N
(b) N = mg * cos(12°) = 5.8 kg * 9.8 m/s² * cos(12°) ≈ 56.03 N
(c) N = mg * cos(25°) = 5.8 kg * 9.8 m/s² * cos(25°) ≈ 50.95 N
(d) N = mg = 5.8 kg * 9.8 m/s² = 56.84 N