An iron bolt of mass 80.0 g hangs from a string 38.8 cm long. The top end of the string is fixed. Without touching it, a magnet attracts the bolt so that it remains stationary, displaced horizontally 28.0 cm to the right from the previously vertical line of the string.
(a) Find the tension in the string.
(b) Find the magnetic force on the bolt.
The string makes an angle of sin^-1 28/38.8 = 46.2 degrees with the vertical.
Do a force balance in vertical and horizontal directions.
T cos 46.2 = Weight = m g
Solve for T, the tension. Then use this equation.
T sin 46.2 = magnetic force
Thanks so much for your help drwls!!
To solve this problem, we can start by finding the tension in the string using the equilibrium conditions.
(a) Find the tension in the string:
In the vertical direction, the tension in the string must balance the weight of the bolt:
Tension in the string = Weight of the bolt
The weight of the bolt can be calculated using the formula:
Weight = mass * gravitational acceleration
Given:
Mass of the bolt (m) = 80.0 g = 0.080 kg
Gravitational acceleration (g) = 9.8 m/s^2 (approximate value)
Weight of the bolt = 0.080 kg * 9.8 m/s^2 = 0.784 N
Therefore, the tension in the string is also 0.784 N.
(b) Find the magnetic force on the bolt:
The magnetic force acting on the bolt is responsible for keeping it in place horizontally.
The magnetic force can be calculated using the formula:
Magnetic force = Force of attraction between the bolt and magnet
Given:
Displacement of the bolt (d) = 28.0 cm = 0.28 m (converted to meters)
Length of the string (l) = 38.8 cm = 0.388 m (converted to meters)
The force of attraction between two magnetic objects is given by:
Force of attraction = (μ₀ * m₁ * m₂) / (4π * r²)
Where:
μ₀ is the permeability constant (4π × 10⁻⁷ T·m/A)
m₁ is the magnetic moment of the magnet (assumed to be 1 for simplicity)
m₂ is the magnetic moment of the bolt (assumed to be 1 for simplicity)
r is the distance between the magnet and the bolt (the horizontal displacement)
The formula can be simplified since m₁ = m₂ = 1:
Force of attraction = (μ₀) / (4π * r²)
Substituting the values:
Force of attraction = (4π × 10⁻⁷ T·m/A) / (4π * 0.28 m)
The 4π terms cancel out:
Force of attraction = (10⁻⁷ T·m/A) / (0.28 m)
The magnetic force on the bolt can be approximated as:
Magnetic force = 3.571 x 10⁻⁷ T·A
Note: The magnetic force is specified in units of teslas (T) times amperes (A) because it is a product of magnetic fields and electrical currents.
To find the tension in the string, we need to consider the forces acting on the bolt.
(a) The tension in the string is equal to the weight of the bolt plus the magnetic force acting on it.
1. Calculate the weight of the bolt:
The weight (W) of an object is given by the formula W = mass × gravitational acceleration (g). In this case, the mass of the bolt is 80.0 g (0.08 kg), and the gravitational acceleration, g, is approximately 9.8 m/s^2. Therefore, the weight of the bolt is W = 0.08 kg × 9.8 m/s^2 = 0.784 N.
2. Calculate the horizontal component of the magnetic force:
The bolt is displaced horizontally by 28.0 cm to the right. Since the bolt is stationary, the magnetic force must be equal in magnitude and opposite in direction to the weight of the bolt. Therefore, the horizontal component of the magnetic force (F_magnetic_horizontal) is 0.784 N.
3. Calculate the vertical component of the magnetic force:
Since the bolt is displaced horizontally but remains stationary, the vertical component of the magnetic force (F_magnetic_vertical) must be equal in magnitude to the tension in the string, T. This is because the sum of the vertical forces acting on the bolt must be zero for the bolt to remain stationary.
4. Find the tension in the string:
To find the tension in the string, we need to find the vertical component of the magnetic force. Since the bolt remains stationary, the vertical component of the magnetic force must balance the weight of the bolt (F_magnetic_vertical = Weight). Therefore, the tension in the string (T) is equal to the weight of the bolt.
So the tension in the string is T = 0.784 N.
(b) To find the magnetic force on the bolt, we already found the horizontal component of the magnetic force, which is 0.784 N. This represents the magnitude of the magnetic force, acting horizontally and to the left.
Therefore, the magnetic force on the bolt is 0.784 N to the left.