An iron bolt of mass 70.0 g hangs from a string 31.3 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 23.0 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
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To find the tension in the string, we need to balance the forces acting on the bolt.
a) Tension in the string:
The only two forces acting on the bolt are its weight (due to gravity) and the tension in the string. Since the bolt is stationary, these forces must be balanced.
1. Weight of the bolt (mg):
The weight of the bolt can be calculated using the formula:
weight = mass x acceleration due to gravity
Given:
mass (m) = 70.0 g = 0.070 kg (converted from grams)
acceleration due to gravity (g) = 9.8 m/s^2
weight (W) = 0.070 kg x 9.8 m/s^2
2. Horizontal force due to tension (T):
The tension in the string is responsible for keeping the bolt in place horizontally. It acts towards the left to balance the horizontal displacement caused by the magnetic force.
Using triangle properties, we can find the vertical component (V) of the tension using the Pythagorean theorem:
V^2 = T^2 - displacement^2
Given:
displacement (d) = 23.0 cm = 0.230 m (converted from centimeters)
string length (L) = 31.3 cm = 0.313 m (converted from centimeters)
V^2 = T^2 - d^2
T^2 = V^2 + d^2
T = √(V^2 + d^2)
To find V, we can use similar triangles:
V / L = d / L
V = (d / L) x L
V = d
Now we can substitute this value of V in T:
T = √(d^2 + d^2)
T = √(2d^2)
T = √2d
Substituting the value of d, we have:
T = √2 x 0.230 m
Now, calculate the value of T.
b) Magnetic force on the bolt:
The magnetic force on the bolt is responsible for keeping it in place horizontally. It acts towards the right and must be equal in magnitude to the horizontal component of the tension in the string.
Given:
displacement (d) = 23.0 cm = 0.230 m (converted from centimeters)
To find the magnetic force (F), we know that it must be equal to the horizontal component of the tension, which is the force responsible for balancing the displacement.
F = T x cos(angle between T and displacement)
Since the bolt is stationary, the angle between the tension and the displacement is 180 degrees. In this case, cos(180 degrees) = -1.
F = T x (-1)
Substitute the value of T obtained in part a), and calculate the value of F.