To determine the magnitude and direction of the force exerted by the hammer claws on the nail, we can break down the given information and solve step-by-step.
Step 1: Find the vertical component of the force exerted by the hammer claws on the nail.
The vertical component of the force can be found using the mass of the hammer, gravitational acceleration, and the angle θ.
Force vertical (Fv) = mass x gravity x sin(θ)
Given:
Mass (m) = 1.00 kg
Gravity (g) = 9.8 m/s²
Angle (θ) = 32.2°
Converting angle from degrees to radians:
θ (in radians) = θ (in degrees) x π / 180
θ (in radians) = 32.2° x π / 180 ≈ 0.562 rad
Substituting the given values to calculate the vertical component of the force:
Fv = 1.00 kg x 9.8 m/s² x sin(0.562 rad)
Fv ≈ 5.37 N
Step 2: Find the horizontal component of the force exerted by the hammer claws on the nail.
The horizontal component of the force can be found using the mass of the hammer, gravitational acceleration, and the angle θ.
Force horizontal (Fh) = mass x gravity x cos(θ)
Using the given values:
Fh = 1.00 kg x 9.8 m/s² x cos(0.562 rad)
Fh ≈ 8.47 N
Step 3: Calculate the perpendicular distance between the contact point and the hammer's force.
Given that the contact point is 30 cm from the force and the head of the nail is 5 cm from the contact point, we can see that the perpendicular distance is the sum of these two distances.
Perpendicular distance (d) = 30 cm + 5 cm
d = 35 cm
Converting the distance to meters:
d = 35 cm x 1 m / 100 cm
d = 0.35 m
Step 4: Calculate the torque exerted by the hammer claws on the nail.
The torque (τ) can be calculated using the following formula:
τ = force x distance
τ = Fh x d
Substituting the given values:
τ = 8.47 N x 0.35 m
τ ≈ 2.96 N·m
Step 5: Determine the magnitude and direction of the force exerted by the hammer claws on the nail.
Since the nail does not move, the force exerted by the hammer claws on the nail must be equal and opposite to the force exerted by the nail on the hammer claws. Therefore, the magnitude of the force exerted by the hammer claws on the nail is 8.47 N.
To find the direction of the force, we need to consider the vertical and horizontal components. The vertical component was found to be 5.37 N, and the horizontal component was found to be 8.47 N.
Using these components, the direction of the force exerted by the hammer claws on the nail can be determined using trigonometry:
θ = arctan(Fv / Fh) = arctan(5.37 N / 8.47 N)
θ ≈ 33.7°
Therefore, the magnitude of the force exerted by the hammer claws on the nail is 8.47 N, and the direction of the force is approximately 33.7° above the horizontal direction.