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(b) Consider a scenario where the block travels in a circular path where the spr...

Mar 30, 2024

(b) Consider a scenario where the block travels in a circular path where the spring is stretched a distance

d

from its unstretched length

L

. i. Determine an expression for the magnitude of the net force

F_{\text {net }}

exerted on the block. Express your answer in terms of

m_{0}, k_{0}, L, d

, and fundamental constants, as appropriate. ii. Derive an equation for the tangential speed

v

of the block. Express your answer in terms of

m_{0}, k_{0}, L, d

, and fundamental constants, as appropriate. (c) Does your equation for the tangential speed

v

of the block from part (b)(ii) agree with your reasoning from part (a)? Yes No Explain your reasoning.

Solution

To determine the net force exerted on the block, we consider the centripetal force required for circular motion, which is provided by the spring force. According to Hooke's Law, the spring force is

F_s = k_0 d

. This force acts as the centripetal force

F_c

, which is required to keep the block in circular motion. Therefore,

F_{\text{net}} = F_c = k_0 d

To derive the equation for the tangential speed

v

of the block, we use the expression for centripetal force

F_c = \frac{m_0 v^2}{r}

. Since the spring force provides the centripetal force, we have

k_0 d = \frac{m_0 v^2}{r}

. Solving for

v

gives us

v = \sqrt{\frac{k_0 d r}{m_0}}

. Here,

r = L + d

, which is the total radius of the circular path. Substituting

r

into the equation for

v

, we get

v = \sqrt{\frac{k_0 d (L + d)}{m_0}}

Answer

The magnitude of the net force exerted on the block is

k_0 d

, and the tangential speed of the block is

\sqrt{\frac{k_0 d (L + d)}{m_0}}

Key Concept

Centripetal Force: Centripetal force is the force that keeps an object moving in a circle. Equation:

F_c = \frac{mv^2}{r}

(Centripetal force equals mass times velocity squared divided by radius)

Explanation

The net force exerted on the block is the spring force acting as the centripetal force, which is necessary for maintaining the block's circular motion. The tangential speed is derived from the balance between the spring force and the centripetal force requirement for circular motion.

Solution

The equation for the tangential speed

v

derived in part (b)(ii) should be consistent with the reasoning from part (a) if the principles of circular motion and Hooke's Law are correctly applied

Answer

Yes

Key Concept

Elasticity and Springs: Hooke's Law states that the force needed to extend or compress a spring by some distance is proportional to that distance. Equation:

F = -kx

(Force equals negative the spring constant times displacement)

Explanation

The equation for the tangential speed

v

agrees with the reasoning from part (a) because it is derived using the correct application of Hooke's Law for the spring force and the formula for centripetal force, which dictates the relationship between the speed of an object in circular motion and the force required to maintain that motion.