An unknown object is evaluated inside a space probe descending towards the surface of a di… — JAMB UTME

An unknown object is evaluated inside a space probe descending towards the surface of a distant planet. A beam balance calibrated on Earth registers a reading of $15.0\text{ kg}$ for the object. The probe has a downward acceleration of $2.0\text{ m s}^{-2}$ in a region where the local gravitational acceleration of the planet is $6.0\text{ m s}^{-2}$ . What reading will a spring balance display when the same object is suspended from it inside the probe?

A
$90.0\text{ N}$
B
$120.0\text{ N}$
$60.0\text{ N}$ Answer
D
$150.0\text{ N}$

Answer

60.0\text{ N}

A beam balance compares gravitational forces on equal balance arms, meaning local gravity and frame acceleration affect both sides equally. Therefore, the beam balance measures the true invariant mass of

15.0\text{ kg}

. A spring balance measures the tension or apparent weight

W_{\text{app}} = m(g - a)

. Substituting

m = 15.0\text{ kg}

, local gravity

g = 6.0\text{ m s}^{-2}

, and downward acceleration

a = 2.0\text{ m s}^{-2}

gives

W_{\text{app}} = 15.0 \times (6.0 - 2.0) = 60.0\text{ N}

Step-by-Step Solution

Determine the true mass of the object using the beam balance measurement

True mass

m = 15.0\text{ kg}

A beam balance compares unknown mass with standard masses under the same local gravitational field. Because local gravity acts equally on both pans, beam balance readings yield the true invariant mass regardless of local gravitational acceleration or uniform reference frame acceleration (provided net effective gravity is greater than zero).

Calculate the effective acceleration experienced inside the accelerating reference frame

g_{\text{eff}} = g_{\text{planet}} - a = 6.0\text{ m s}^{-2} - 2.0\text{ m s}^{-2} = 4.0\text{ m s}^{-2}

When a reference frame accelerates downward at rate

a

, the effective apparent gravitational acceleration felt by objects suspended inside is reduced by

a

Calculate the apparent weight registered by the spring balance

W_{\text{app}} = m \cdot g_{\text{eff}} = 15.0\text{ kg} \times 4.0\text{ m s}^{-2} = 60.0\text{ N}

A spring balance measures tension force (apparent weight) exerted on its spring, which equals

m(g - a)

in a downward accelerating system.

Key Concept

Mass measurement via beam balance vs. apparent weight measurement via spring balance in accelerating frames

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